Ch 03_ Motion in a Plane 2

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Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane
CONCEPT: INTRO TO MOTION IN 2D ● Motion at an angle (2D : A→C ) is just combining TWO straight-line motions (1D : A→B & B→C ), with vector equations . - Whenever motion is in 2D, FIRST break it down into X & Y (1D) . MOTION EQs VECTOR EQs 𝒗 𝒂𝒗𝒈 = 𝚫? 𝚫? 𝒂 𝒂𝒗𝒈 = 𝚫𝒗 𝚫𝐭 UAM (1) 𝒗 = 𝒗 ? + 𝒂? (2) 𝒗 ? = 𝒗 ? ? + ?𝒂𝚫? (3) 𝚫? = 𝒗 ? ? + ? ? 𝒂? ? (4)* 𝚫? = (𝒗 ? +𝒗) ? ? ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝐴 ⃗ 𝐴 ? 𝐴 ? 𝜃 ? +? +? ? ? ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 1
CONCEPT: POSITION AND DISPLACEMENT IN 2D ● If Position & Displacement are 2D vectors, use vector equations to jump between 2D vector ⇔ X & Y components . |? ሬ⃗| = ඥ ? ? + ? ? ? ሬ⃗ = ? cos θ 𝜽 = tan -1 ቀ |?| |?| ቁ ? ሬ⃗ = ? sin θ |𝚫? ሬሬሬሬሬ⃗ | = ඥ 𝚫? ? + 𝚫? ? 𝚫? ሬ⃗ = 𝚫? cos θ 𝜽 = tan -1 ቀ |𝚫?| |𝚫?| ቁ 𝚫? ሬ⃗ = 𝚫? sin θ O A B A B Position _________ ? ሬ⃗ ⇔ Components +x +y O Displacement Vector 𝚫? ሬ⃗ ⇔ Components +x +y - ______________ in position - Where you are / Coordinate (_____) Arrow from ___________ → Point Shortest path from Point → Point EXAMPLE 1 : At point A , your position is 3.6m @ 33.7° . You move to point B , where your position is 8.49m @ 45° . Calculate the x & y components of your position at A & B . EXAMPLE 2 : Using Example 1, calculate the magnitude & direction of the displacement from A to B . Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 2
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PROBLEM : At point A , a hiker is 10m east from the origin . After 35s, the hiker arrives at point B 40m at 60° north of east from the origin . Calculate the magnitude and direction of the hiker ’s displacement . A) 50m ; 60° north of east B) 36m ; 73.9° north of east C) 36m ; 60° north of east D) 36m ; 16.1°north of east PROBLEM : Your initial position is 6.2 m from the origin, 25° below the x-axis . You then travel 9.9 m at an angle 78° above the positive x-axis, then 2.0 m in the negative x-direction . What is the magnitude & direction of your final position vector? A) 13.5m ; 58° above +x-axis B) 18.1m ; 78° above +x-axis C) 9.06m ; 51.2° above +x-axis D) 10.4m ; 42.6° above the x-axis |? ሬ⃗| = ඥ ? ? + ? ? ? ሬ⃗ = ? cos θ 𝜽 = tan -1 ቀ |?| |?| ቁ ? ሬ⃗ = ? sin θ |𝚫? ሬሬሬሬሬ⃗ | = ඥ 𝚫? ? + 𝚫? ? 𝚫? ሬ⃗ = 𝚫? cos θ 𝜽 = tan -1 ቀ |𝚫?| |𝚫?| ቁ 𝚫? ሬ⃗ = 𝚫? sin θ 2D POSITION / DISPLACEMENT EQUATIONS |? ሬ⃗| = ඥ ? ? + ? ? ? ሬ⃗ = ? cos θ 𝜽 = tan -1 ቀ |?| |?| ቁ ? ሬ⃗ = ? sin θ |𝚫? ሬሬሬሬሬ⃗ | = ඥ 𝚫? ? + 𝚫? ? 𝚫? ሬ⃗ = 𝚫? cos θ 𝜽 = tan -1 ቀ |𝚫?| |𝚫?| ቁ 𝚫? ሬ⃗ = 𝚫? sin θ 2D POSITION / DISPLACEMENT EQUATIONS Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 3
𝜽 𝒗 = ______ CONCEPT: AVERAGE SPEED AND VELOCITY IN 2D ● Remember : Average speed & velocity measure how FAST something moves between two points . EXAMPLE : You walk 40m in the +x-axis, then 30m in the +y-axis . The entire trip takes 10 seconds . Calculate a) your average speed b) the magnitude & direction of your velocity ? = ?𝐢??𝐚??? ?𝐢?? ⇒ 𝒅 𝚫? Speed (Magnitude only) |𝒗 ⃗ 𝒂𝒗𝒈 | = ?𝐢?𝐩?𝐚?????? ?𝐢?? ⇒ 𝚫? +x +y - [ SCALAR | VECTOR ]; always points in same direction as ____ Velocity (Magnitude + Direction) - [ SCALAR | VECTOR ] Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 4
PROBLEM : While following a treasure map, you start at an old oak tree . You first walk 85 m at 30.0° west of north, then walk 92 m at 67 . 0° north of east . You reach the treasure 2 minutes later . Calculate the magnitude of your average velocity for the entire trip . A) 1.11 m/s B) 1.48 m/s C) 1.40 m/s D) 1.32 m/s PROBLEM : While following a treasure map, you start at an old oak tree . You first walk 85 m at 30.0° west of north, then walk 92 m at 67 . 0° north of east . You reach the treasure 2 minutes later . Calculate your average speed for the entire trip . A) 1.5 m/s B) 177m/s C) 88.5 m/s |𝒗 ⃗ 𝒂𝒗𝒈 | ⇒ 𝚫? 𝚫? 𝑠 ⇒ ? 𝚫? 2D SPEED / VELOCITY EQUATIONS |𝒗 ⃗ 𝒂𝒗𝒈 | ⇒ 𝚫? 𝚫? 𝑠 ⇒ ? 𝚫? 2D SPEED / VELOCITY EQUATIONS Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 5
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CONCEPT: CALCULATING VELOCITY COMPONENTS ● If 𝒗 ⃗ 𝒂𝒗𝒈 is 2D, it has x & y components . There are 2 sets of equations to go back & forth between 𝒗 ⃗ 𝒂𝒗𝒈 & components : 1) Velocity Components ↔ Displacement & Time 2) Velocity Components ↔ Magnitude & Direction 𝚫? = 40 𝚫? = 30 |𝒗 ⃗ | = 𝚫? 𝚫? = ξ 𝒗 ? ⃗⃗⃗⃗ = ______ = ___ cos θ 𝜽 𝒗 = tan -1 ቀ | | | | ቁ 𝒗 ? ⃗⃗⃗ = ______ = ___ sin θ 𝒗 ⃗ = 5 𝜽 = 37° EXAMPLE : You walk 40 m right, then 30 m up in 10s . Calculate the velocity’s magnitude and its x & y components . EXAMPLE : You walk at 5m/s at an angle 37° above the x-axis . Calculate the x & y components of your velocity . Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 6
PROBLEM : A coastal breeze pushes your sailboat at constant velocity for 8 min . After checking your instruments, you determine you’ve been pushed 650 m west and 800 m south . What was the magnitude & direction of your average velocity? A) 2.15 m/s ; 39.1° south of west B) 128.9 m/s ; 50.9° south of west C) 2.15 m/s ; 50.9° south of west PROBLEM : A ball moves on a tabletop . The ball has initial x & y coordinates (1.8m, 3.6m) . The ball moves 10m/s at 53.1° above the x-axis for 4s . What are the x & y coordinates of the ball’s final position? A) (11.8m,13.6m) B) (25.8m, 35.6m) C) (41.8m, 43.6m) D) (33.8m, 27.6m) 2D Velocity Vector |𝒗 ⃗ | = 𝚫? 𝚫? = ඥ 𝑣 ? 2 + 𝑣 ? 2 𝒗 ? ⃗⃗⃗⃗ = 𝚫? 𝚫? = 𝒗 cos θ 𝜽 𝒗 = tan -1 ൬ | 𝐯 ? | | 𝐯 ? | ൰ 𝒗 ? ⃗⃗⃗ = 𝚫? 𝚫? = 𝒗 sin θ 2D Velocity Vector |𝒗 ⃗ | = 𝚫? 𝚫? = ඥ 𝑣 ? 2 + 𝑣 ? 2 𝒗 ? ⃗⃗⃗⃗ = 𝚫? 𝚫? = 𝒗 cos θ 𝜽 𝒗 = tan -1 ൬ | 𝐯 ? | | 𝐯 ? | ൰ 𝒗 ? ⃗⃗⃗ = 𝚫? 𝚫? = 𝒗 sin θ Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 7
CONCEPT: ACCELERATION IN 2D ● Remember ! Acceleration (in 1D & 2D) causes a change in _____________ ( magnitude and/or direction ) . - Just like velocity, there are two sets of equations to calculate acceleration and its components : EXAMPLE : A toy car is initially moving 20m/s in the +x-axis . 10 seconds later, the car is moving 67 m/s at 26.5° above the x-axis . a) Calculate the x & y components of the car’s acceleration . b) Calculate the magnitude & direction of the car’s acceleration over the 10s . 𝜃 𝑣 𝒗 ሬԦ Velocity 𝜃 𝑎 𝒂 ሬԦ Acceleration | 𝒗 | = 𝚫? 𝚫? = ට𝒗 ? 𝟐 + 𝒗 ? 𝟐 𝜽 𝒗 = tan -1 ቀቚ 𝐯 ? 𝐯 ? ቚቁ 𝒗 ? = 𝚫? 𝚫? = 𝒗 𝒄??𝜽 𝒗 ? = 𝚫? 𝚫? = 𝒗 ?𝒊?𝜽 | 𝒂 | = 𝚫𝒗 𝚫? = ξ 𝜽 𝒂 = tan -1 ቀቚ ቚቁ 𝒂 ? = 𝚫? = ____ 𝒄??𝜽 𝒂 ? = 𝚫? = ____ ?𝒊?𝜽 Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 8
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PROBLEM : A football at rest is kicked by a football kicker . The ball is in contact with the kicker’s foot for 0.050s, during which it experiences an acceleration 𝑎 = 340 m/s 2 . The ball is launched at an angle of 40° above the ground (x-axis) . Calculate the horizontal and vertical components of the launch velocity . A) 13 m/s horizontal ; 10.9 m/s vertical B) 130 m/s horizontal ; 109 m/s vertical C) 10.9 m/s horizontal ; 13 m/s vertical D) 17 m/s horizontal ; 17 m/s vertical 2D Acceleration Vector |𝒂 ሬԦ| = 𝚫𝒗 𝚫? = ඥ 𝑎 ? 2 + 𝑎 ? 2 𝒂 ? ሬሬሬሬԦ = 𝚫𝒗 ? 𝚫? = 𝒂 cos θ 𝜽 𝒂 = tan -1 ൬ ห 𝐚 ? ห | 𝐚 ? | ൰ 𝒂 ? ሬሬሬԦ = 𝚫𝐯 ? 𝚫? = 𝒂 sin θ Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 9
CONCEPT: SOLVING KINEMATICS PROBLEMS IN 2D ● Solving constant acceleration problems in 2D is done the same way as in 1D! - Remember: Separate 2D motion into two 1D motions and solve . EXAMPLE : A hockey puck slides along a lake at 8m/s east . A strong wind accelerates the puck at a constant 3 m/s 2 in a direction 37° northeast . What is the magnitude & direction of the hockey puck’s displacement after 5s ? 2D MOTION w/ ACCELERATION 1) Draw Diagram & decompose vectors into x & y 2) List 5 variables for x & y , identify known & target variables 3) Pick UAM Eq . without “Ignored” Variable 4) Solve UAM Equations X Y (1) 𝒗 ? = 𝒗 ?? + 𝒂 ? ? (1) 𝒗 ? = 𝒗 ?? + 𝒂 ? ? (2) 𝒗 ? ? = 𝒗 ?? ? + ?𝒂 ? 𝜟? (2) 𝒗 ? ? = 𝒗 ?? ? + ?𝒂 ? 𝜟? (3) 𝜟? = 𝒗 ?? ? + ? ? 𝒂 ? ? ? (3) 𝜟? = 𝒗 ?? ? + ? ? 𝒂 ? ? ? (4) 𝜟? = ( 𝒗 ? +𝒗 ?? ? ) ? (4) 𝜟? = ( 𝒗 ? +𝒗 ?? ? ) ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 10
PROBLEM : A survey drone has just completed a scan at x,y coordinates (57m, 8m) at t=0 . It needs to return to a lab located at (-115, 72) m . If its initial velocity is 16m/s in the +y-direction, and it has only 18s of battery life remaining, what constant acceleration (magnitude and direction) does it need to reach the lab? A) 2.8 m/s 2 ; along – x axis B) 1.8 m/s 2 ; 51.8° below – x axis C) 2.7 m/s 2 ; above – x axis D) 1.3 m/s 2 ; 24° above – x axis 2D MOTION w/ ACCELERATION 1) Draw Diagram & decompose vectors into x & y 2) List 5 variables for x & y , identify known & target variables 3) Pick UAM Eq . without “Ignored” Variable 4) Solve MOTION EQs VECTOR EQs 𝒗 𝒂𝒗𝒈 = 𝚫? 𝚫? 𝒂 𝒂𝒗𝒈 = 𝚫𝒗 𝚫𝐭 UAM (1) 𝒗 = 𝒗 ? + 𝒂? (2) 𝒗 ? = 𝒗 ? ? + ?𝒂𝚫? (3) 𝚫? = 𝒗 ? ? + ? ? 𝒂? ? (4)* 𝚫? = (𝒗 ? +𝒗) ? ? 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄??(𝜽 ? ) 𝑨 ? = 𝑨 ?𝒊?(𝜽 ? ) 𝐴 ⃗ 𝐴 ? 𝐴 ? 𝜃 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 11
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CONCEPT: INTRODUCTION TO PROJECTILE MOTION ● Projectile Motion occurs when an object is launched & moves in 2D under the influence of only ___________ . - Remember ! Whenever we have Physics problems in 2D, we decompose them into 1D (X & Y) . ● Projectile Motion COMBINES (1) horizontal motion where 𝑎 ? = ___, and (2) vertical motion where 𝑎 ? = ____ . VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) UAM EQUATIONS X (a x = 0) Y (a y = – g) (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 ? )𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 ? )𝐭 HORIZONTAL LAUNCH DOWNWARD LAUNCH UPWARD LAUNCH EQUATIONS TO USE FOR PROJECTILE MOTION VERTICAL MOTION 1D 2D PROJECTILE MOTION 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 12
PROBLEM : Which of the following quantities are constant during projectile motion? A) Vertical acceleration & vertical velocity B) Angle (direction) of the velocity vector C) Horizontal acceleration & vertical velocity D) Vertical acceleration & horizontal velocity Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 13
CONCEPT: SOLVING PROJECTILE MOTION PROBLEMS (WITH HORIZONTAL LAUNCH EXAMPLE) EXAMPLE : A ball rolls horizontally off a 2m-tall table with a speed of 3.0 m/s . Calculate a) the time it takes for the ball to hit the ground , and b) the horizontal displacement (range) of the ball ● In projectile motion, time t can be found from either X OR Y axis equations . Always try the ___ axis equation first . - If you get stuck and can't solve using X axis equation, always try to solve it with a Y axis equation, and vice versa . ● When an object is launched horizontally , its initial velocity is ONLY in the ___ axis : - Remember: all objects in projectile motion always have (1) 𝒂 ? = 0, so 𝒗 ? ________ changes ; and (2) 𝒂 ? = – g PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝒗 ?? = ____ 𝒗 ?? = ____ = ____ 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 14
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PROBLEM : A rock is thrown horizontally with a speed of 20 m/s from the edge of a high cliff . It lands 80 m from the cliff's base . How tall is the cliff? A) 78.4 m B) 19.6 m C) 122.5 m D) 24.5 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 15
PROBLEM : A ping-pong player standing 1.6 m from the net serves the ball horizontally . The ball is hit 1.2 m above the floor . What initial speed does the ball need to go over the net, which is 1.6m away from the player and 0.90m above the floor? A) 2.1 m/s B) 3.2 m/s C) 9.2 m/s D) 6.4 m/s PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 16
PROBLEM : You kick a ball horizontally at 8 m/s from the roof of a 40m-tall building . Unfortunately, a car below you on the street accelerates forwards from rest, and your ball lands on the car . What was the acceleration of the car? A) 7.92 m/s 2 B) 5.60 m/s 2 C) 1.96 m/s 2 D) 11.2 m/s 2 PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 17
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CONCEPT: SOLVING DOWNWARD LAUNCH PROBLEMS ● When an object is launched downward , v 0y will always be [ POSITIVE | NEGATIVE ]. EXAMPLE : You throw a rock at 5m/s angled 37° downward from a building . It hits the ground 10m from the building . Calculate a) the height of the building and b) the magnitude & direction of its velocity just before hitting the ground . ● Remember! If you get stuck and can't solve using X axis equation, try to solve it with a Y axis equation, and vice versa . PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 18
PROBLEM : Water pours from a spout at the end of a gutter with a speed of 3.5 m/s, where the spout is angled 45° downwards . The magnitude of the water’s velocity when it hits the ground is 14 m/s . How high is the spout from the ground? A) 18.8 m B) 10.6 m C) 10 m D) 9.4 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 19
PROBLEM : A cannon mounted on a tall fort fires a cannonball with 73 m/s at 49° below the horizontal . If the fort is 300m above the ground, what horizontal distance does the cannonball travel before hitting the ground? A) 292 m B) 192 m C) 220 m D) 374.4 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 20
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CONCEPT: SOLVING SYMMETRICAL LAUNCH PROBLEMS ● If an object is launched upwards , v 0y is always [ POSITIVE | NEGATIVE ] - The maximum height or peak is always a point of interest, because 𝑣 (𝑝𝑒𝑎𝑘),? = ___ ● If an object returns to the _________________ from which it was launched (___ = ___), its trajectory is symmetrical. - For symmetrical launches ONLY : & & & EXAMPLE : You kick a football at 20m/s angled 53° upward, and it later returns to the ground . Calculate a) the time the football takes to reach its max height ; b) the total time of flight ; c) the vertical component of the football’s velocity when it returns to the ground . PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) ____ = ____ ____ = ____ ____ = ____ ____ = ____ 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 21
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PROBLEM : A flare gun launches signal flares with an initial speed of 110 m/s . How far does the flare travel if it is shot at ground level at an angle 65° above the horizontal? A) 1890 m B) 944 m C) 507 m D) 1040 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 22
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PROBLEM : In a game of catch on a faraway planet, a ball is thrown with 10 m/s at 37° above the horizontal . It travels a horizontal distance of 32 m and lands on the ground . What is the magnitude of the gravitational acceleration on this planet? A) 0.3 m/s 2 B) 1.5 m/s 2 C) 3 m/s 2 D) 6 m/s 2 PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄?𝒔(𝜽 ? ) 𝑨 ? = 𝑨 𝒔𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 23
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CONCEPT: SOLVING NON-SYMMETRICAL UPWARD LAUNCH PROBLEMS ● IF an object is launched upward and lands at a HIGHER or LOWER point, the motion is non-symmetrical. ● IF landing at a LOWER height, part of motion (A→C) is symmetrical, but object drops further (C→D) . - When choosing intervals in problems, try to include point ___ (max height) to simplify equations, because 𝒗 ?? = ___ EXAMPLE : You fire a potato from a launcher on a 20m-high cliff . The potato has an initial speed of 30m/s at 53° upwards . The potato reaches its max height 49.4m above the ground after 2.45s . Find th e vertical component of the potato’s velocity just before hitting the ground . PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? LANDS AT HIGHER HEIGHT (Launch TO Height) LANDS AT LOWER HEIGHT (Launch FROM Height) LANDS AT SAME HEIGHT (Symmetrical Launch) Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 24
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PROBLEM : You throw a rock off the top of a tall building at an upward angle of 15° . At t=3 s, the rock ’s horizontal displacement from you is 52m . How high does the rock get above the top of the building? A) 1.1 m B) 4.6 m C) 30 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 25
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PROBLEM : A child throws a ball from ground level with an initial speed of 13 m/s at an upward angle of 67.4° . It reaches its maximum height directly above the edge of a roof, then lands on the roof, 3 m from the edge . How high is the roof? A) 7.3 m B) 2.9 m C) 5.6 m D) 4.4 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 26
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PROBLEM : A catapult launches a stone with an initial velocity of 50 m/s at an angle of 56° above the horizontal . What is the direction of the stone's velocity when it hits a castle wall 6 seconds later? A) – 31.7° B) 31.7° C) – 47.7° PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 27
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CONCEPT: USING SINGLE INTERVALS IN UPWARD LAUNCH PROBLEMS ● In projectile motion, you can often choose different intervals (A→B, B→D, etc…) and still get the right answer! - Usually you should try to solve these problems using a single interval (___ → ___) because it’s better/simpler/faster! EXAMPLE : You fire a cannon with 100 m/s at 30° above the +x-axis from a 40-m cliff . Find a) the vertical component of the velocity at point when the cannonball hits the ground, and b) the total time of flight PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) A C B D A C B D 𝜟? ?𝑫 = ___________ OR ____________ 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 28
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PROBLEM : A ball is thrown from the top of a 50-m-tall building with a speed of 40m/s at an angle of 37° above the horizontal . How far horizontally does the ball travel before hitting the ground? A) 101.2 m B) 50.6 m C) 207 m D) 414 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 29
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PROBLEM : A ball is thrown from the top of a 100-m-tall building at an angle of 37° above the horizontal . 3 s later, it breaks a window at a lower height in a building 24 m away . How high above the ground is the window? A) 26 m B) 52 m C) 24 m D) 62 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs ? = √? ? ? + ? ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |? ? | |? ? | ) ? ? = ? 𝒄??(𝜽 ? ) ? ? = ? ?𝒊?(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 30
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CONCEPT: RELEASING OR LAUNCHING PROJECTILES FROM MOVING VEHICLES ● You’ll need to solve problems where projectiles are launched from vehicles already moving with velocity (_____) . - IF a projectile is simply dropped/released, then the moving vehicle and projectile move at the ______ velocity . ● ? ???? is the ___________________ of the launch velocity and the velocity it BORROWS from the moving vehicle . EXAMPLE : A cart carrying a vertical missile launcher moves horizontally at a constant 60m/s to the right . The missile launches vertically upward at 80 m/s . What is the maximum height achieved by the rocket? PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄??(𝜽 ? ) 𝑨 ? = 𝑨 ???(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? ? ???? = _____________ PROJECTILE DROPPED / RELEASED PROJECTILE LAUNCHED / THROWN ? ?𝒆? = 300 m/s ? ?𝒆? = -3 m/s ? 𝒍𝒂??𝒄? = 4m/s ? ?𝒆? = 30 m/s ? 𝒍𝒂??𝒄? = 40 m/s Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 31
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PROBLEM : A small plane flies horizontally at 20m/s at an altitude of 200m, when it launches a projectile at a speed of 65 m/s at 22.6° below the horizontal . What horizontal distance does the projectile travel before hitting the ground? A) 752 m B) 344 m C) 184 m D) 920 m PROJECTILE MOTION 1) Draw paths in X&Y and points of interest (Points of Interest : initial, final, max height, etc.) 2) Determine target variable 3) Determine interval and UAM equation 4) Solve UAM EQUATIONS X Y 𝚫? = 𝐯 ? 𝐭 (1) 𝐯 ? = 𝐯 ?? + 𝐚 ? 𝐭 (2) 𝐯 ? ? = 𝐯 ?? ? + ?𝐚 ? 𝚫? (3) 𝚫? = 𝐯 ?? 𝐭 + ? ? 𝐚 ? 𝐭 ? *(4) 𝚫? = ? ? (𝐯 ?? + 𝐯 𝐟 )𝐭 VECTOR EQs 𝑨 = √𝑨 ? ? + 𝑨 ? ? 𝜽 ? = 𝐭𝐚𝐧 −? ( |𝑨 ? | |𝑨 ? | ) 𝑨 ? = 𝑨 𝒄??(𝜽 ? ) 𝑨 ? = 𝑨 ???(𝜽 ? ) 𝜃 ? 𝐴 Ԧ 𝐴 ? 𝐴 ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 32
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CONCEPT: USING SPECIAL EQUATIONS IN SYMMETRICAL LAUNCH PROBLEMS ● For symmetrical launches ONLY ( ? ? = ? 𝟎 ), remember : - In addition, there are even more special equations you may be allowed to use! EXAMPLE : A catapult launches a projectile with 100m/s at 53° upwards . The projectile later returns to the ground . Find : a) the time the projectile hits the ground b) horizontal range of the projectile c) the other launch angle that gives the same range A B C 𝑹 ??? = Δ? 𝑨𝑪 = 𝒗 𝟎 𝟐 ?𝒊?(𝟐𝜽) ? TOTAL HORIZONTAL DISPLACEMENT (aka RANGE) (RANGE) TOTAL TIME OF FLIGHT - MAXIMUM when launch angle θ = ____ . - Complementary angles (i.e. 𝜃 1 + 𝜃 2 = ____, e.g ____& ____) achieve same 𝑹 for same 𝒗 𝟎 . ? ??? = 𝚫? 𝑨𝑪 = 𝟐𝒗 𝟎 ?𝒊?𝜽 ? ? ↑ = ? ↓ 𝒗 ↑ = −𝒗 ↓ |𝒗 𝒄 | = |𝒗 𝒂 | 𝜽 𝒄 = – 𝜽 𝒂 If you have ? 𝑨𝑪 : If you don’t have ? 𝑨𝑪 : 𝑹 ??? = Δ? 𝑨𝑪 = 𝒗 ? ? 𝑨𝑪 +y +x Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 33
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PROBLEM : A frog leaps with a speed of 15 m/s and lands on the ground 2.0s later . At what angle did the frog jump? A) 40.8° B) 9.4° C) 19.1° Symmetrical Launches ? ↑ = ? ↓ 𝒗 ↑ = −𝒗 ↓ |𝒗 𝒄 | = | 𝒗 𝒂 | 𝜽 𝑪 = −𝜽 𝑨 ? 𝐬𝐲𝐦 = 𝚫? 𝑨𝑪 = 𝟐𝒗 𝟎 ?𝒊?𝜽 ? 𝑹 ??? = 𝚫? 𝐀𝐂 = 𝒗 𝟎 𝟐 ?𝒊?(𝟐𝜽) ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 34
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PROBLEM : A champion long-jumper competing on Planet X is capable of leaving the ground with a speed of 6 m/s . The maximum distance he can cover on Planet X turns out to be 9 m . What is the gravitational acceleration on Planet X? A) 0.75 m/s 2 B) 4.0 m/s 2 C) 2.1 m/s 2 D) 12 m/s 2 Symmetrical Launches ? ↑ = ? ↓ 𝒗 ↑ = −𝒗 ↓ |𝒗 𝒄 | = | 𝒗 𝒂 | 𝜽 𝑪 = −𝜽 𝑨 ? 𝐬𝐲𝐦 = 𝚫? 𝑨𝑪 = 𝟐𝒗 𝟎 ?𝒊?𝜽 ? 𝑹 ??? = 𝚫? 𝐀𝐂 = 𝒗 𝟎 𝟐 ?𝒊?(𝟐𝜽) ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 35
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PROBLEM : A golf ball is hit at ground level at an angle of 31.9° above the horizontal . Its range is 257 m over a level green . What was the magnitude of the golf ball's initial velocity? A) 2807 m/s B) 69 m/s C) 95 m/s D) 53 m/s SPECIAL EQUATIONS ? 𝐬𝐲𝐦 = 𝚫? 𝑨𝑪 = 𝟐𝒗 𝟎 ?𝒊?𝜽 ? 𝑹 ??? = 𝚫? 𝐀𝐂 = 𝒗 𝟎 𝟐 ?𝒊?(𝟐𝜽) ? Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 36
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CONCEPT: INTRO TO UNIFORM CIRCULAR MOTION ● In Uniform Circular Motion (UCM), objects move in a circular path with __________________ . - 𝒗 ⃗ changes direction in UCM ; 𝒗 ⃗ at any point is called the ____________ velocity ( 𝒗 ⃗ 𝑻 ) . - 𝒂 ⃗ (“centripetal” = center -seeking) points towards _________ of the path ( 𝒂 ⃗ 𝑪 or 𝒂 ⃗ 𝒓𝒂? ) . - 𝑹 is the distance from the edge of the path to the center, or the ___________ of the path . EXAMPLE : You move at constant 5 m/s when you turn into a circle of radius 10m . Calculate your centripetal acceleration . 𝒂 ? = Units : [ _____ ] Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 37
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PROBLEM : A ball travels on a frictionless circular track at 3m/s . The ball cannot have an acceleration greater than 1.5m/s 2 or it will go off the track . What is the smallest radius the circular track can have so that the ball stays on the track? PROBLEM : The Moon travels in a circular orbit of radius R = 3.85×10 8 m around the Earth because of gravity . Because of the large distance, the centripetal acceleration of the Moon is only 0.0026m/s 2 . How fast would the Moon be moving if it suddenly broke free of Earth’s gravity and stopped orbiting? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 𝟐 𝑹 Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 𝟐 𝑹 Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 38
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CONCEPT: CIRCUMFERENCE, PERIOD AND FREQUENCY IN UNIFORM CIRCULAR MOTION ● When objects complete a full ROTATION a.k.a. REVOLUTION a.k.a. CYCLE: - the distance traveled is called the CIRCUMFERENCE ___ = _____ - Period (__) → # of __________ per __________ ; Unit : [ __ ] or [ ] - Frequency (__) → # of __________ per __________ ; Unit : [ ___ = ] or [ ] EXAMPLE : Calculate the period and frequency of your motion if you complete : PROBLEM : Modern windmills usually spin at a rate of 20 Revolutions Per Minute ( R.P.M ) . At this rate, how long does it take for a windmill blade to complete a full rotation? ● Whenever you’re given Revs per Minute (RPMs), to get frequency → 𝒇 = 𝐑𝐏𝐌 𝟔? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? T = a) 4 rotations in 2 seconds b) 0.5 rotations in 3 seconds 𝒇 = ⇔ Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 39
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CONCEPT: MORE EQUATIONS FOR VELOCITY AND ACCELERATION IN UNIFORM CIRCULAR MOTION ● We can combine the circumference, period, and frequency into equations for 𝒗 𝑻 and 𝒂 𝑪 : EXAMPLE : A ball moves in a circle of radius 10m . Calculate : 𝒗 𝑻 = = OR _________ a) its speed if it takes 60 seconds to complete 100 rotations 𝒂 𝑪 = OR ___________ 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 ⇒ 𝒗 𝑻 = distance time ⇒ 𝑹 b) its centripetal acceleration if it completes 1 rotation every 3 minutes Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 40
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PROBLEM : A 3kg rock spins horizontally at the end of a 2m string at 90 RPM . Calculate its centripetal acceleration . PROBLEM : A big problem for astronauts in space is the lack of gravity ! One way to simulate gravity is to build a space ship with spinning rings attached to it . If a cylindrical space station of diameter = 500m is spun about its axis, how fast in revolutions per minute (RPM) must it turn so the astronauts inside feel an acceleration equal to that of Earth ( g) ? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Young & Adams - 11th edition - College Physics Ch 03: Motion in a Plane Page 41
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