Principles of Physics: A Calculus-Based Text, Hybrid (with Enhanced WebAssign Printed Access Card)
Principles of Physics: A Calculus-Based Text, Hybrid (with Enhanced WebAssign Printed Access Card)
5th Edition
ISBN: 9781305586871
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 3, Problem 19P

A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of vi= 18.0 m/s. The cliff is h = 50.0 m above a body of water as shown in Figure P3.19. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land?

Chapter 3, Problem 19P, A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed

(a)

Expert Solution
Check Mark
To determine

The coordinates of the initial position of the stone .

Answer to Problem 19P

The coordinates of the initial position of the stone are xi=0.00m,yi=0.00m_.

Explanation of Solution

Write the expression for the initial position of the stone.

    ri=xi+yi        (I)

Here, ri is the initial position of the stone, xi is the horizontal initial position of the stone and yi is the vertical initial position of the stone.

Conclusion:

Substitute 0.00m for xi and 0.00m for yi in (I) to find ri

    ri=0.00m+0.00m

Therefore, the coordinates of the initial position of the stone are xi=0.00m,yi=0.00m_.

(b)

Expert Solution
Check Mark
To determine

The components of the initial velocity of the stone .

Answer to Problem 19P

The components of the initial velocity of the stone are vxi=18.0m/s,vyi=0.0m/s_.

Explanation of Solution

Write the expression for the initial velocity of the stone,

    vi=vxi+vyi        (II)

Here, vi is the initial velocity of the stone, vxi is the horizontal component of the initial velocity and vyi is the vertical component of the initial velocity of the stone.

Conclusion:

Substitute 18.0m/s for vxi and 0.0m/s for vyi in (II) to find vi ,

    vi=18.0m/s+0.0m/s

Therefore, the components of the initial velocity of the stone are vxi=18.0m/s,vyi=0.0m/s_.

(c)

Expert Solution
Check Mark
To determine

The vertical motion of the stone .

Answer to Problem 19P

The vertical motion of the stone is 9.8m/s2_.

Explanation of Solution

In this case, the vertical motion of the stone is equal to the free fall motion.

It is with a constant downward acceleration.

    ay=g        (III)

Here, ay is the vertical downward motion and g is the acceleration due to gravity.

Conclusion:

Substitute 9.8m/s2 for g in (III) to find ay,

    ay=9.8m/s2

Therefore, the vertical motion of the stone is 9.8m/s2_.

(d)

Expert Solution
Check Mark
To determine

The horizontal motion of the stone .

Answer to Problem 19P

The horizontal motion of the stone is constantmotion_.

Explanation of Solution

In this case, the constant velocity motion in the horizontal direction.

Conclusion:

Thus, there is no horizontal acceleration from gravity.

Therefore, the horizontal motion of the stone is constantmotion_.

(e)

Expert Solution
Check Mark
To determine

The symbolic equations for the xandy components of the velocity of the stone as a function of time .

Answer to Problem 19P

The symbolic equations for the xandy components of the velocity of the stone as a function of time are vxf=vxiandvyf=gt_.

Explanation of Solution

Write the expression for the horizontal final velocity of the stone.

    vxf=vxi+axt        (IV)

Here, vxf is the horizontal final velocity of the stone, vxi is the horizontal initial velocity of the stone, ax is the horizontal acceleration component of the stone and t is the flight time.

Write the expression for the vertical final velocity of the stone.

    vyf=vyi+ayt        (V)

Here, vyf is the vertical final velocity of the stone, vyi is the vertical initial velocity of the stone, ay is the vertical acceleration component of the stone and t is the flight time.

Conclusion:

Substitute 0 for ax in (IV) to get vxf ,

    vxf=vxi

Substitute 0 for vyi and g for ay in V) to get vyf ,

    vyf=gt

Therefore, the symbolic equations for the xandy components of the velocity of the stone as a function of time are vxf=vxiandvyf=gt_.

(f)

Expert Solution
Check Mark
To determine

The symbolic equations for the position of the stone as a function of time .

Answer to Problem 19P

The symbolic equations for the position of the stone as a function of time are xf=vxitandyf=12gt2_.

Explanation of Solution

Write the expression for the horizontal final position of the stone.

    xf=xi+vxit+12axt2        (VI)

Here, xf is the horizontal final position of the stone, xi is the horizontal initial position of the stone, vxi is the horizontal initial velocity of the stone, ax is the horizontal acceleration component of the stone and t is the flight time.

Write the expression for the vertical final position of the stone.

    yf=yi+vyit+12ayt2        (VII)

Here, yf is the vertical final position of the stone, yi is the vertical final position of the stone, vyi is the vertical initial velocity of the stone, ay is the vertical acceleration component of the stone and t is the flight time.

Conclusion:

Substitute 0 for ax and 0 for xi  in (VI) to get xf ,

    xf=vxit

Substitute 0 for yi, vyi for 0 and g for ay in (VII) to get yf ,

    yf=12gt2

Therefore, the symbolic equations for the xandy components of the velocity of the stone as a function of time are xf=vxitandyf=12gt2_.

(g)

Expert Solution
Check Mark
To determine

The time of impact of the stone .

Answer to Problem 19P

The time of impact of the stone is t=3.19s_.

Explanation of Solution

Write the expression for the vertical final position of the stone.

    yf=12gt2        (VIII)

Here, yf is the vertical final position of the stone, g is the vertical acceleration of the stone and t is the flight time.

Rewrite the above equation,

    h=12gt2t=2hg        (IX)

Conclusion:

Substitute 9.80m/s2 for g and 50.0m for h  in (IX) to get t ,

    t=2(50.0m)9.80m/s2=3.19s

Therefore, the time of impact of the stone is t=3.19s_.

(h)

Expert Solution
Check Mark
To determine

The speed and angle of impact of the stone land .

Answer to Problem 19P

The speed and angle of impact of the stone land are vf=36.1m/sandθ=60.1o_.

Explanation of Solution

In this case, at the time of impact vxf=vxi .

The vertical component velocity of the stone, vyf=gt .

Substitute 2hg for t in the above equation,

    vyf=g2hg=2gh        (X)

Write the expression for the final velocity of the stone.

    vf=vxf2+vyf2        (XI)

Here, vf is the final velocity of the stone

Write the expression for the angle of impact of the stone.

    θ=tan1vyfvxf        (XII)

Here, θ is the angle of impact of the stone.

Conclusion:

Substitute 9.8m/s2 for g and 50.0m for h  in (X) to get vyf ,

    vyf=2(9.8m/s2)(50.0m)=31.3m/s .

Substitute 18.0m/s2 for vxf and 31.3m/s for vyf in (XI) to get vf ,

    vf=(18.0m/s2)2+(31.3m/s)2=36.1m/s .

Substitute 31.3m/s for vyf and 18.0m/s for vxf in (XII) to find θf ,

    θf=tan131.318.0=60.1o

Therefore, the speed and angle of impact of the stone land are vf=36.1m/sandθ=60.1o_.

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Chapter 3 Solutions

Principles of Physics: A Calculus-Based Text, Hybrid (with Enhanced WebAssign Printed Access Card)

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