Question

At the top of a raised platform, a ball is thrown vertically upward with a speed of 60 m/s. The ball rises up, then falls to the ground below the platform with a downward speed of 90 m/s.(a) For how much time was the ball in free-fall?(b) How far is the platform from the ground?

Answer 1

Before we begin to answer the question we need to notice that since the gravitaional acceleration will act on the motion this is a uniform accelerated motion. This means that we can use the following equations:

$g=\frac{v_f-v_0}{t}$$y=y_0+v_0t+\frac{1}{2}gt^2$$v^2_f-v^2_0=2g(y-y_0)$

Before we begin we establish that upward is the positve direction, this means that g=-10 m/s^2 and that the velocity of the ball when it reaches the ground below the platform is -90 m/s. We also establis that the origin of motion is at the platform

a)

To determine the time the ball is free-falling we need to know the maximum height the ball reaches before it begins falling.

-Motion from the platform to the maximum height.

In this case the initial velocity is 60 m/s while the final velocity is 0 m/s; furthermore the initial height is zero since our origin is on the platform.

The height it reaches the ball before it starts to fall can be obtained by the third equation:

$\begin{gathered} 0^2-60^2=2(-10)(y-0) \\ -3600=-20y \\ y=-\frac{3600}{20} \\ y=180 \end{gathered}$

Therefore the maximum height is 180 meters.

Now the time it takes the ball to reaches this height is given by the first formula:

$\begin{gathered} -10=\frac{0-60}{t} \\ t=\frac{-60}{-10} \\ t=6 \end{gathered}$

Hence it takes the ball six seconds to reach its maximum height.

Now, we know that in an accelerated uniform motion the time it takes to reach the maximum height is the same as the tame it takes to reach the initial height, therefore it takes 6 seconds for the ball to travel from the maximum height to the platform again.

Now we need to determine the time it takes the ball to fall from the height of the platform to the bottom; in this case the initial velocity is -60 m/s; this comes from the fact that the final velocity is the same at same heights in this kind of motion. Then, using the first formula we know that the time it takes is:

$\begin{gathered} -10=\frac{-90-(-60)}{t} \\ t=\frac{-30}{-10} \\ t=3 \end{gathered}$

Hence, it takes 3 seconds for the ball to travel from the platform to the ground.

Finally we add the three times to determine the total time of the free fall; therefore the time the ball is free falling is 15 seconds.

b)

To determine the height of the platform we can use the fact that the time it takes the ball to fall from this height to the ground is six seconds, then using the second equation we have that:

$\begin{gathered} y=0-60(3)+\frac{1}{2}(-10)(3)^2 \\ y=-225 \end{gathered}$

This means that the ball travels 225 meters downward from the height of the platform to the ground, therefore the platform is 225 meters above the ground.