# Two baseballs are thrown off the top of a building that is 7.24 m high. Both are thrown with...

## Question:

Two baseballs are thrown off the top of a building that is 7.24 m high. Both are thrown with initial speed of 63.3 mph. Ball 1 is thrown horizontally, and ball 2 is thrown straight down. What is the difference in the speeds of the two balls when they touch the ground? (Neglect air resistance.)

## Conservation of Energy:

This problem can be solved in a general way using projectile motion, but that will be a huge mess. We will solve the problem using the conservation of energy. The conservation of energy says that if no external force is acting on a system then the total energy of that system remains constant. Using this consideration we will solve this problem.

In this problem, two kinds of energy are involved. One is the kinetic energy of the ball and the other is the potential energy of the balls due to the height.

Let us assume a ball has mass {eq}m {/eq} and the initial velocity is {eq}v_i {/eq} and is at height {eq}h {/eq}

Then the kinetic energy of the ball is {eq}K_i=\dfrac12mv_i^2 {/eq} and potential energy is {eq}P_i=mgh {/eq}.

Let us assume when it reaches the ground it has velocity {eq}v_f {/eq} and since we assumed the ground as a reference, at ground potential energy will be zero.

Therefore from conservation of energy, final total energy= initial total energy.

\begin{align} &K_f+P_f=K_i+P_i\\[.3 cm] &\dfrac12mv_f^2+0=\dfrac12mv_i^2+mgh\\[.3 cm] &\dfrac {v_f^2}2=\dfrac {v_i^2}2+gh\\[.3 cm] &v_f^2=v_i^2+2gh\\[.3 cm] &v_f=\sqrt{v_i^2+2gh} \end{align}

Now, we can see that final speed is independent of mass and angle of projection.

Since both balls were thrown with the same initial speed and the same height, therefore, the final speed of both balls will be the same. Hence the difference of the speed of the balls when it reaches the ground is zero.