Question

A mother has four times the mass of her young son. Both are running with the same kinetic energy. What is the ratio v_son /v_mother of their speeds?

Short answer

The ratio of speed is v_son /v_{mother =} 2:1

Step-by-step solution

Given information

A mother has four times the mass of her young son.

Both are running with the same kinetic energy

Explanation

The kinetic energy is calculated by

KE = 1/2(mv²) .....................................(1)

Use equation (1) to calculate the kinetic energy of son and mother and find the ratio

Lets assume

Suppose K . E_son is the kinetic energy for son and

K . E_mother is the kinetic energy for mother and it is given that both have same kinetic energy

$$\begin{gathered} K·E_{\text{son}}=K·E_{\text{mother}} \\ 0.5\times M_{\text{son}}\times V_{\text{son}}{}^2=0.5\times M_{\text{mother}}\times V_{\text{mother}}^2 \\ {M}_{\text{mother}}=4\times M_{\text{son}} \\ \left(V_{\text{son}}^2/V_{\text{mother}}^2\right)=M_{\text{mother}}/M_{\text{son}} \\ \text{Substitute the given values} \\ {\left(V_{\text{son}}/V_{\text{mother}}\right)}^2=4/1 \\ \text{Take square root both sides} \\ \left(V_{\text{son}}/V_{\text{mother}}\right)=2/1=2:1 \end{gathered}$$