Momentum and Kinetic Energy Relationship

Understanding the relationship between momentum and kinetic energy is fundamental in physics. When the momentum of a body changes, its kinetic energy also changes, provided the mass remains constant. Let's explore how kinetic energy is affected when the momentum of a body is doubled.

Defining Momentum and Kinetic Energy

• Momentum (p): Momentum is a measure of the mass in motion. It is defined as the product of an object's mass (m) and its velocity (v).

  \[p = mv\]

• Kinetic Energy (KE): Kinetic energy is the energy an object possesses due to its motion. It is defined as one-half of the product of the object's mass (m) and the square of its velocity (v).

  \[KE = \frac{1}{2}mv^2\]

Deriving the Relationship Between Kinetic Energy and Momentum

We can establish a direct relationship between kinetic energy and momentum. From the momentum formula, we can express velocity (v) in terms of momentum (p) and mass (m):

\[v = \frac{p}{m}\]

Now, substitute this expression for velocity into the kinetic energy formula:

\[KE = \frac{1}{2}m\left(\frac{p}{m}\right)^2\]

Simplifying the expression:

\[KE = \frac{1}{2}m\frac{p^2}{m^2}\]

\[KE = \frac{p^2}{2m}\]

This derived formula, \(KE = \frac{p^2}{2m}\), clearly shows that kinetic energy is directly proportional to the square of the momentum (\(p^2\)) and inversely proportional to the mass (m).

Calculating Kinetic Energy When Momentum is Doubled

Let's consider the initial state of the body:

• Initial momentum = \(p_1\)
• Initial kinetic energy = \(KE_1\)
• Mass = \(m\) (which remains unchanged)

From our derived formula:

\[KE_1 = \frac{p_1^2}{2m}\]

Now, the problem states that the momentum of the body is doubled. So, the new momentum (\(p_2\)) is:

\[p_2 = 2p_1\]

Let's find the new kinetic energy (\(KE_2\)) using the relationship \(KE = \frac{p^2}{2m}\) with the new momentum:

\[KE_2 = \frac{p_2^2}{2m}\]

Substitute \(p_2 = 2p_1\) into the equation:

\[KE_2 = \frac{(2p_1)^2}{2m}\]

Simplify the expression:

\[KE_2 = \frac{4p_1^2}{2m}\]

We can rearrange this to see the relationship with the initial kinetic energy:

\[KE_2 = 4 \left(\frac{p_1^2}{2m}\right)\]

Since \(KE_1 = \frac{p_1^2}{2m}\), we can conclude that:

\[KE_2 = 4 \times KE_1\]

Therefore, when the momentum of a given body is doubled and its mass remains unchanged, its kinetic energy becomes four times its original value.