To determine how kinetic energy changes when linear momentum changes, we need to understand the relationship between these two quantities.

Understanding Momentum and Kinetic Energy

The linear momentum ($p$) of an object is given by the product of its mass ($m$) and velocity ($v$):

$ p = mv $

The kinetic energy ($KE$) of an object is given by:

$ KE = \frac{1}{2}mv^2 $

We can express kinetic energy in terms of momentum. From the momentum equation, $v = \frac{p}{m}$. Substituting this into the kinetic energy equation gives:

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

Calculating Kinetic Energy Change Factor

Let the initial momentum be $p_1$ and the initial kinetic energy be $KE_1$. Then:

$ KE_1 = \frac{p_1^2}{2m} $

The problem states that the linear momentum changes by two times. This means the new momentum, $p_2$, is twice the initial momentum:

$ p_2 = 2p_1 $

The new kinetic energy, $KE_2$, corresponding to the new momentum $p_2$ is:

$ KE_2 = \frac{p_2^2}{2m} $

Substitute $p_2 = 2p_1$ into the equation for $KE_2$:

$ KE_2 = \frac{(2p_1)^2}{2m} = \frac{4p_1^2}{2m} $

To find the factor by which the kinetic energy changes, we calculate the ratio $\frac{KE_2}{KE_1}$:

$ \frac{KE_2}{KE_1} = \frac{\frac{4p_1^2}{2m}}{\frac{p_1^2}{2m}} $

Simplifying this expression:

$ \frac{KE_2}{KE_1} = \frac{4p_1^2}{2m} \times \frac{2m}{p_1^2} = 4 $

Conclusion

Therefore, if the linear momentum of a moving object changes by two times, its kinetic energy will change by a factor of 4.