   Chapter 3, Problem 10P

Chapter
Section
Textbook Problem

A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range R. (a) At what angle θ is the rock thrown? (b) In terms of the original range R, what is the range Rmax the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

(a)

To determine
The angle with which the rock is thrown so that the maximum height of the rock s equal to its horizontal range.

Explanation

The following diagram shows the projectile motion of rock thrown upward.

At maximum height, the vertical component of the velocity is zero. So, the kinematic equation becomes

vy=v0y+ayt0=v0y+ayt

• v0y is the vertical component of the launch speed
• ay is the acceleration in the vertical direction
• t is the time taken

The time to reach the maximum height is,

tp=0v0yay

Substitute v0sinθ for v0y and g for ay .

tp=v0sinθg=v0sinθg

The maximum vertical displacement is,

hmax=(vy)avt=(0+v0y2)tp

Substitute v0sinθ for v0y and (v0sinθg) for tp .

hmax=(v0sinθ2)(v0sinθg)=v02sin2θ2g

Now, the horizontal range is the product of the horizontal component of the launch velocity and the total time

(b)

To determine
The maximum range in terms of R when the rock is thrown with the same speed but at an angle optimal for maximum range.

(c)

To determine
If the angle with which the rock is thrown so that the maximum height of the rock is equal to its horizontal range will be different on a different planet.

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