Animated Solution for Physics - Kinematics: A ball whose kinetic energy is $E$, is projected at an angle of $45^{\circ }$ to the horizontal. The kinetic energy of the ball at the highest point of its flight will be

Launching the Projectile

Imagine you are standing on an open field, and you throw a ball into the air at an angle of $45^{\circ }$ to the horizontal. The ball possesses an initial kinetic energy, which we will call $E$. This energy is born from the initial velocity $u$ you imparted to the ball, mathematically expressed as $E=\frac{1}{2}m{u}^2$.

As the ball travels, it traces a beautiful parabolic path. Gravity constantly pulls it downward, slowing its ascent, but what happens to its forward motion? Let's break it down.

A Moment of Horizontal Purity

When the ball reaches the absolute highest point of its flight, something magical happens. For a brief, infinitesimal moment, the ball stops moving upwards. Its vertical velocity, $u_y$, becomes exactly zero.

However, gravity only acts vertically. There is no force pushing or pulling the ball horizontally (assuming we ignore air resistance). Because of this, the horizontal component of the velocity, $u_x$, remains completely unchanged throughout the entire flight.

At the peak, the ball's entire velocity is purely horizontal. We can calculate this using trigonometry:

Since our launch angle $\theta$ is $45^{\circ }$, we substitute this in:

Calculating the Remaining Energy

Now that we know the velocity at the highest point is $\frac{u}{\sqrt{2}}$, we can find the new kinetic energy, $E^{\prime }$. We simply plug this new velocity into our standard kinetic energy formula:

Substituting our value for $u_x$:

Let's carefully square the term inside the parentheses. The square of $u$ is $u^2$, and the square of $\sqrt{2}$ is $2$. This gives us:

To see how this relates to our initial energy, let's rearrange the terms slightly. We can pull the $\frac{1}{2}$ out to the front:

Look closely at the term inside the parentheses: $\frac{1}{2}m{u}^2$. That is exactly our initial kinetic energy, $E$!

Therefore, we can substitute $E$ back into the equation:

At the highest point of its trajectory, the ball retains exactly half of its initial kinetic energy. The other half has been temporarily converted into gravitational potential energy. This elegant result highlights the beautiful symmetry and conservation laws governing projectile motion.