Chapter 4, Problem 54A

If you throw a ball straight upward at a speed of 10 m/s, how long will it take to reach zero speed? How long will it take to return to its starting point? How fast will it be going when it returns to its starting point?

To determine

The time taken by the ball to reach zero speed, time taken to return to its starting point, and the speed of the ball when it returns to its starting point.

Answer to Problem 54A

The time taken by the ball to reach zero speed is $1.02\text{ s}$ .

Time taken to return to its starting point is $2.04\text{ s}$ .

The speed of the ball when it returns to its starting point is

  $10.0\text{ m/s}$ .

Explanation of Solution

Given:

The ball is thrown straight upward with velocity (initial velocity of the ball) $u=10\text{ m/s}$

Formula used:

The time taken by the ball to reach zero speed can be calculated by using the equation,

  $v=u+at$

  $\Rightarrow at=v-u$

  $\Rightarrow t=\frac{v-u}{a}$ $\left(1\right)$

Where, $v$ is the final velocity of the ball

  $a$ is the acceleration of the ball, since the ball is moving in upward direction,

  $a=-g=-9.8{\text{ m/s}}^2$

The speed of the ball when it returns to its starting point can be calculated by using the equation,

  $v=u+at$ $\left(2\right)$

Where, $u$ is the initial velocity of the ball that is $u=0.0\text{ m/s}$

  $a$ is the acceleration of the ball, since the ball is moving in downward direction,

  $a=g=9.8{\text{ m/s}}^2$

Substituting for $u$ and $a$ in equation $\left(2\right)$ ,

  $v=\left(9.8{\text{ m/s}}^2\right)t$ $\left(3\right)$

Calculation:

The time taken by the ball to reach zero speed can be calculated by substituting the numerical values in equation $\left(1\right)$ ,

  $t=\frac{0-10\text{ m/s}}{-9.8{\text{ m/s}}^2}=1.02\text{ s}$

Since the ball has taken $1.02\text{ s}$ to reach its highest point, same time is required to the ball to return from this highest point to its starting point. Hence the total time required to the ball to return to its starting point is

  $\text{2t = 2}\left(1.02\text{ s}\right)=2.04\text{ s}$

The speed of the ball when it returns to its starting point can be calculated by substituting the numerical values in equation $\left(2\right)$ ,

  $v=\left(9.8{\text{ m/s}}^2\right)\left(1.02\text{ s}\right)=9.996\text{ m/s}\approx 10.0\text{ m/s}$

Conclusion:

The time taken by the ball to reach zero speed is $1.02\text{ s}$ .

Time taken to return to its starting point is $2.04\text{ s}$ .

The speed of the ball when it returns to its starting point is $10.0\text{ m/s}$ .