Question

a car of mass 1250 kg and a bus of mass 5000 kg are moving with eqaul linear momentum. find the ratio of their kinetic energies

Answer 1

By definition, the linear momentum of a body is:

$$p = mv$$

where $m$ is the mass of a body, and $v$ is its velocity. Since both cars have equal linear momentum, we can write:

$$p_1 = p_2\\ m_1v_1 = m_2v_2$$

where $m_1 = 1250kg, m_2 = 5000kg$ are the masses, and $v_1, v_2$ are the velocities of the first and second cars respectively. Thus, obtain the ratio:

$$\dfrac{v_1}{v_2} = \dfrac{m_2}{m_1}\\ \space \\ \dfrac{v_1^2}{v_2^2} = \dfrac{m_2^2}{m_1^2}$$

On the other hand, their kinetic energies are given as follows:

$$K_1 = \dfrac{m_1v_1^2}{2}\\ K_2 = \dfrac{m_2v_2^2}{2}$$

Their ratio is:

$$\dfrac{K_1}{K_2} = \dfrac{m_1v_1^2}{m_2v_2^2}$$

Substituting the expression for $v_1^2/v_2^2$, obtain:

$$\dfrac{K_1}{K_2} = \dfrac{m_1\cdot m_2^2}{m_2\cdot m_1^2} = \dfrac{m_2}{m_1}\\ \dfrac{K_1}{K_2} =\dfrac{5000kg}{1250kg} = 4$$

Answer. 4.