Animated Solution for Physics - Work, Energy, and Power: From a building, two balls $A$ and $B$ are thrown such that $A$ is thrown upwards and $B$ downwards (both vertically). If $v_A$ and $v_B$ are their respective velocities on reaching the ground, then

Two Balls, Two Directions

Imagine you are standing at the edge of a tall building of height $h$. You have two balls, A and B, in your hands. You throw ball A straight up into the sky with an initial speed $u$. At the exact same moment, you throw ball B straight down towards the ground with the same initial speed $u$.

The question asks us to compare their final velocities, $v_A$ and $v_B$, right before they crash into the ground. At first glance, you might think ball A will hit the ground faster because it falls from a greater height after reaching its peak. Or maybe ball B hits faster because it was thrown directly downwards? Let's find out using the magic of physics.

Conservation of Energy

To solve this elegantly, we will use the Principle of Conservation of Mechanical Energy. Since we are ignoring air resistance, the total mechanical energy (Potential Energy + Kinetic Energy) of each ball remains constant throughout its entire journey.

Let's write the energy equation for ball A. At the moment it leaves your hand, it has a gravitational potential energy of $m_{A}gh$ and a kinetic energy of $\frac{1}{2}{m}_A{u}^2$.

When it finally reaches the ground, its height is zero, so its potential energy vanishes. All of its initial energy has been converted into kinetic energy, which is $\frac{1}{2}{m}_A{v}_A^2$.

Equating the initial and final energies, we get:

Notice something beautiful here? The mass $m_A$ is present in every term! We can completely cancel it out. This proves a fundamental truth: the final velocity of an object falling under gravity is completely independent of its mass.

Rearranging the equation to solve for $v_A$, we get:

Now, let's analyze ball B. It was thrown downwards, but does that change its initial energy? No! Kinetic energy is a scalar quantity. It depends on the square of the speed ($u^2$), not the direction of the velocity vector.

Therefore, the initial kinetic energy of ball B is also $\frac{1}{2}{m}_B{u}^2$, and its initial potential energy is $m_{B}gh$. Setting up the exact same energy conservation equation for ball B yields:

Comparing the two final expressions, the result is crystal clear. Both balls hit the ground with the exact same speed!

Even though ball A took a longer trip by going up first, by the time it falls back down to the level of the building's roof, it will be traveling downwards with the exact same speed $u$ that ball B started with. From that point on, their journeys are identical.