Answer to Question #161974 in Classical Mechanics for Gill Allain

Question

A particle of mass m moves with momentum of magnitude p

A). Show that the kinetic energy of the particle is K = p^2 / 2m.

B). An object has a kinetic energy of 275J and a momentum of magnitude 25.0 kgm/s. Find the speed and mass of the object.

Answer 1

A) By the definition of the linear momentum, we have:

$$p=mv.$$

Then, we can write the kinetic energy of the particle as follows:

$$K=\dfrac{1}{2}mv^2=\dfrac{1}{2}\dfrac{m\cdot mv^2}{m}=\dfrac{1}{2}\dfrac{(mv)^2}{m}=\dfrac{p^2}{2m}.$$

B) Let's firs find the mass of the object from the formula:

$$K=\dfrac{p^2}{2m},$$ $$m=\dfrac{p^2}{2K}=\dfrac{(25.0\ \dfrac{kgm}{s})^2}{2\cdot275\ J}=1.14\ kg.$$

Finally, we can find the speed of the object from the definition of the linear momentum:

$$v=\dfrac{p}{m}=\dfrac{25.0\ \dfrac{kgm}{s}}{1.14\ kg}=22\ \dfrac{m}{s}.$$