03_Projectile Motion_LAB (Spring 2024 Online Version)

1Projectile Motion – Background and Theory Objective In this laboratory activity, students will observe the horizontal range of a projectile shot from various heights and angles. In addition, students will compare the time of flight for projectiles shot horizontally at different initial velocities. Theory Gravitational Acceleration and Free Fall In this experiment, we will observe objects that are solely under the influence of gravitational acceleration, or Free Fall . An object is considered to be in free fall if it is in the air, and not under any external force except for gravity. We will be ignoring the effect of air resistance for this experiment. Free fall does not necessarily mean that an object is moving downwards – when an object is launched upward , as we will see in this experiment, it is considered to be in free fall the moment it loses contact with the external force that is propelling it. One example is throwing a ball, where the moment the ball leaves your hand it can be considered to be in free fall, even if it is thrown directly upward. An object that is in free fall is only under the influence of gravitational force, and so it has a constant acceleration due to gravity: a y =− g≈ − 9.8 m s 2 (Eq. 1) The negative sign is used here to represent that the acceleration is acting downward . This is the only acceleration that the object experiences during free fall, so we can additionally state that a x = 0. If air resistance is ignored, this is true independently of the object’s mass, size or shape! In a vacuum it can be shown that a bowling ball and a feather will fall with the same acceleration, as shown in figure 1. Projectile Motion In order to study the 2-dimensional motion of objects in free fall, it is useful to separate the horizontal and vertical components of motion. This leads to 2 sets of kinematic formulas, which are connected only by their time variable (t) but are otherwise mathematically independent. Figure 1 – Free fall of a bowling ball vs. feathers. When they are dropped in the absence of air resistance, they undergo the same motion. (animation available here )

Projectile Motion Page 2 Horizontal Vertical ⃑ v fx = ⃑ v 0 x + ⃑ a x t ⃑ v fy = ⃑ v 0 y + ⃑ a y t Δ ⃑ x = 1 2 ( ⃑ v fx + ⃑ v 0 x ) t Δ ⃑ y = 1 2 ( ⃑ v fy + ⃑ v 0 y ) t Δ ⃑ x = ⃑ v 0 x t + 1 2 ⃑ a x t 2 Δ ⃑ y = ⃑ v 0 y t + 1 2 ⃑ a y t 2 ⃑ v fx 2 = ⃑ v 0 x 2 + 2 ⃑ a x Δ ⃑ x ⃑ v fy 2 = ⃑ v 0 y 2 + 2 ⃑ a y Δ ⃑ y By inserting our acceleration values for free fall (a x and a y ), the horizontal range,  x, for a projectile can be found using the following equation:  x = v x t (Eq. 2) where v x is the horizontal velocity (the initial horizontal velocity) and t is the time of flight. To find the time of flight, t , the following kinematic equation is needed:  y = v 0y t + ½ a y t 2 (Eq. 3) where  y = y – y o is the vertical displacement, a y = g is the acceleration due to gravity and v y0 is the vertical component of the initial velocity. When a projectile is fired horizontally from its initial position y o = 0,  y < 0, and the time of flight can be found by rearranging Equation 2. Since the initial vertical velocity is zero, the last term drops out of the Equation 2, yielding: t = (2  y/a y ) 1/2 = (-2  y/g) 1/2 (Eq. 3a) When a projectile is fired at an angle and it lands at the same elevation from which it was launched,  y = 0, and we may solve Equation (2) for t: t = 2v 0y /g (Eq. 3b) Substituting this into Equation (1) yields Δ x = 2 v x v 0 y g = 2 v 0 2 cos θ sin θ g = v 0 2 sin ( 2 θ ) g (Eq. 4) where v is the initial speed of the projectile. When a projectile is fired from a height, none of the terms drop out and Equation 3 may be rearranged as follows: Updated: Fall 2022

Projectile Motion Page 4 Section: _____________ Name: ____________________ Projectile Motion – Lab Report GOAL: (briefly state what experiment(s) will be performed and with what purpose) PROCEDURE 1: Initial Velocity, Time of Flight, and Range In this procedure you will be simulating an object fired horizontally. The website you will be using is: https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html 1. Navigate to the website, into the “Intro” tab. 2. Use the following initial conditions for the simulation:  Height = 10 m  Initial Velocity = 20 m/s  Angle = 0 degrees  Air resistance is turned on (checked)  Acceleration Vectors, Components is turned on (checked)  Speed is set to “Slow” 3. Fire the object using the red button at the bottom of the screen 4. Use the blue tool in the toolbox and inspect the time of flight and range of the pumpkin once it has hit the ground Updated: Fall 2022

Projectile Motion Page 5 5. Repeat this process to fill the values in Table 1 below Table 1: Angle (degrees) Height (m) Initial Velocity (m/s) Time of Flight (s) Range (m) 0 10 20 0 10 18 0 10 16 0 10 14 0 10 12 0 10 10 0 10 8 0 10 6 0 10 4 0 10 2 0 10 0 Use Equation 3a from the theory section to calculate the time of flight: t = ________ s Compare the time of flight values from table 1 (experimental values) with the calculated time of flight (theoretical value) by calculating the percent error . Percent Error = | Theoretical Value - Experimental Value Theoretical Value | ∗ 100% Time of Flight Comparison from Table 1: Initial Velocity from Table 1 (m/s) Time of Flight from Table 1 (s) Percent Error (%) 10 20 0 Updated: Fall 2022

Projectile Motion Page 7 PROCEDURE 2: Launching at an Angle on a Plane In this procedure, you will launch the object at various angles on a level plane. This means that the object will begin and end its motion at the same height (displacement  y = 0) 1. Navigate to the “Lab” tab at the bottom of the page 2. Use the following initial settings  Angle: 25 degrees  Initial Velocity: 15 m/s  Air resistance is on (checked)  Keep all default settings for cannonball object and gravity 3. Use the tools to measure the range of the cannonball and input the information into “Measured Range” in Table 2. 4. Use Eq. 4 from theory to fill in to calculate the Predicted range for the 25-degree angle 5. Repeat for the angles 35-85 degrees and fill in Table 2 Table 2: Initial Velocity (m/s) Angle , q (degrees) Sin(2 q ) Predicted Range (m) [v 0 2 sin(2 q )/g] Measured Range (m) % Error 15 25 15 35 15 45 15 55 15 65 15 75 Updated: Fall 2022

Projectile Motion Page 8 15 85 Based on the data in Table 2, calculate the average percent error between the predicted (PRange) and the measured range (MRange): _________ % What angle corresponds to the maximum range? Why is this so? (confirm with the equation) CONCLUSION (Part II) Evaluate your results and discuss the difference between the predicted and actual range of the object. Is there a way to remove this difference? PROBLEM SOLVING (Show complete solutions for full credit) 1. A tennis player returning a ball hits it with a horizontal speed of 50 m/s at a height of 50 cm. a) How long does it take the ball to reach the ground? b) How far horizontally will the ball land? 2. A soccer ball was kicked with an initial velocity of 15 m/s at an angle of 30 degrees with respect to the horizontal. a) How far horizontally will the ball land? b) What is the total flight time? Updated: Fall 2022