Question

A particle of mass m is initially at rest. A time varying force $F=kt$ acts on it. Its velocity after time t is

• A. $\frac{kt^{2}}{2m}$
• B. $\frac{kt^{2}}{m}$
• C. $\frac{2kt^{2}}{m}$
• D. $\frac{k^{2}t^{2}}{m}$
• E. $\frac{2k^{2}t^{2}}{m}$

Hint:

When force is constant, we use $v = u + at$. But when force depends on time, we MUST use integration. The area under a Force-Time graph divided by mass gives the change in velocity.

Answer 1

The Correct Option is A

Concept:
Physics - Newton's Second Law and Integration.
Force $F = m \cdot a = m \cdot \frac{dv}{dt}$.
Step 1: Set up the differential equation.
Given $F = kt$ and $F = m\frac{dv}{dt}$: $$ kt = m\frac{dv}{dt} $$
Step 2: Separate variables and integrate.
$$ dv = \frac{k}{m}t \, dt $$ Integrate both sides from initial rest ($v=0, t=0$) to final state ($v, t$): $$ \int_{0}^{v} dv = \int_{0}^{t} \frac{k}{m}t \, dt $$
Step 3: Solve the integral.
$$ [v]_{0}^{v} = \frac{k}{m} \left[ \frac{t^2}{2} \right]_{0}^{t} $$ $$ v = \frac{kt^2}{2m} $$