Understanding Kinetic Energy When Projecting a Particle Upwards

When a particle is projected upwards into the air, its motion is primarily affected by the force of gravity, which acts downwards. Let's analyze how this affects the particle's kinetic energy.

Kinetic energy is the energy an object possesses due to its motion. It is directly related to the mass and speed of the object. The formula for kinetic energy ($KE$) is given by:

\begin{equation*} KE = \frac{1}{2}mv^2 \end{equation*}

Where:

• $m$ is the mass of the particle
• $v$ is the speed (magnitude of velocity) of the particle

When the particle is projected upwards, it starts with an initial upward velocity. As it rises, the gravitational force pulls it downwards, opposing its upward motion. This downward force causes the particle to slow down. In other words, the speed ($v$) of the particle decreases as it moves higher.

Since kinetic energy is proportional to the square of the speed ($KE \propto v^2$), if the speed ($v$) decreases while the mass ($m$) remains constant, the kinetic energy ($KE$) must also decrease.

Let's consider the journey of the particle:

• Just after projection: The particle has maximum upward speed and therefore maximum kinetic energy.
• As it rises: Gravity acts against the motion, causing the speed to decrease. Consequently, the kinetic energy decreases.
• At the highest point: The particle momentarily stops before starting to fall. At this point, its speed is zero ($v=0$). According to the kinetic energy formula, when $v=0$, $KE = \frac{1}{2}m(0)^2 = 0$. So, the kinetic energy is zero at the peak of the trajectory.
• As it falls back down: Gravity now acts in the direction of motion, causing the speed to increase. As the speed increases, the kinetic energy increases.

Therefore, when a particle is projected upwards, its kinetic energy decreases as it rises against the force of gravity.

Revision Table: Kinetic Energy of Upward Projected Particle

Additional Information on Energy in Upward Projection

While the kinetic energy of the particle projected upwards decreases, its potential energy increases. Potential energy is the energy stored by an object due to its position, specifically its height in this case. As the particle goes higher, its gravitational potential energy ($PE$) increases according to the formula:

\begin{equation*} PE = mgh \end{equation*}

Where:

• $m$ is the mass of the particle
• $g$ is the acceleration due to gravity
• $h$ is the height above a reference point

In the absence of air resistance, the total mechanical energy (sum of kinetic energy and potential energy) of the particle remains constant throughout its flight. This is known as the conservation of mechanical energy.

As the particle moves upwards, kinetic energy is transformed into potential energy. The decrease in kinetic energy is equal to the increase in potential energy, keeping the total energy constant.

At the highest point, all the initial kinetic energy has been converted into gravitational potential energy (relative to the projection point).