Question

Two objects having masses in \(1:4\) ratio are at rest. When both of them are subjected to the same force separately, they achieved the same kinetic energy during times \(t_1\) and \(t_2\) respectively. The ratio of \(\frac{t_2}{t_1}\) is

• A. \(4\)
• B. \(2\)
• C. \(2.5\)
• D. \(1\)

Hint:

In problems involving forces, masses, and kinetic energy, always consider how mass and force influence acceleration and how this impacts other physical quantities like velocity and time to achieve certain energy states.

Answer 1

The Correct Option is B

Assuming the masses are \(m\) and \(4m\) and they are subjected to the same force \(F\), the acceleration for each will be \(a_1 = \frac{F}{m}\) and \(a_2 = \frac{F}{4m}\) respectively. The kinetic energy \(K\) achieved by each is given by the expression: \[ K = \frac{1}{2}mv^2 = \frac{1}{2}m(at)^2 \] Setting the kinetic energies equal for both times: \[ \frac{1}{2}m(a_1 t_1)^2 = \frac{1}{2}(4m)(a_2 t_2)^2 \] \[ m\left(\frac{F}{m} t_1\right)^2 = 4m\left(\frac{F}{4m} t_2\right)^2 \] \[ \left(\frac{F}{m} t_1\right)^2 = 4\left(\frac{F}{4m} t_2\right)^2 \] \[ t_1^2 = 4 \times \left(\frac{t_2^2}{4}\right) \] \[ t_1^2 = t_2^2 \] Thus, since \(t_2 = 2 t_1\), the ratio \(\frac{t_2}{t_1}\) is \(2\).