Kinetic Energy and Momentum Relationship

Let's understand the relationship between kinetic energy and momentum to determine which object has more kinetic energy when they have the same momentum but different masses.

We are given two objects, A and B.

• Object A has mass $m_A$.
• Object B has mass $m_B$.

We are also given that the mass of A is more than the mass of B:

\(m_A > m_B\)

Both objects have the same momentum. Let the momentum be \(p\).

\(p_A = p_B = p\)

Deriving Kinetic Energy from Momentum

The formula for momentum (\(p\)) is mass (\(m\)) times velocity (\(v\)):

\(p = mv\)

The formula for kinetic energy (\(KE\)) is half times mass times velocity squared:

\(KE = \frac{1}{2}mv^2\)

We can express velocity from the momentum formula as \(v = \frac{p}{m}\). Substituting this into the kinetic energy formula gives us kinetic energy in terms of momentum and mass:

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

So, kinetic energy is given by:

\(KE = \frac{p^2}{2m}\)

Comparing Kinetic Energies of Objects A and B

Now we apply this relationship to objects A and B. Since their momentum is the same (\(p_A = p_B = p\)), their kinetic energies are:

\(KE_A = \frac{p^2}{2m_A}\)

\(KE_B = \frac{p^2}{2m_B}\)

We are given that \(m_A > m_B\). To compare \(KE_A\) and \(KE_B\), we compare the terms \(\frac{1}{2m_A}\) and \(\frac{1}{2m_B}\), as \(p^2\) is the same for both.

Since \(m_A > m_B\), the denominator \(2m_A\) is greater than \(2m_B\). When the denominator is larger, the fraction is smaller (assuming the numerator is positive). Therefore:

\(\frac{1}{2m_A} < \frac{1}{2m_B}\)

Multiplying both sides by \(p^2\) (which is positive), we get:

\(\frac{p^2}{2m_A} < \frac{p^2}{2m_B}\)

This means:

\(KE_A < KE_B\)

Conclusion on Kinetic Energy

Object B, which has less mass, will have more kinetic energy than object A, which has more mass, when both objects have the same momentum.

Therefore, object B will have more kinetic energy.