A vehicle of mass $m$ is moving on a rough horizontal road with momentum $p$. If the coefficient of friction b

Question

A vehicle of mass $m$ is moving on a rough horizontal road with momentum $p$. If the coefficient of friction between the tyres and the road be $\mu$, then the stopping distance is:

Accepted Answer

1. **Kinetic Energy and Momentum:** The kinetic energy ($K$) of a body can be expressed in terms of its momentum ($p$) and mass ($m$) as: 
   $$ K = \frac{p^2}{2m} $$
   [Source 57, 78].
2. **Work Done by Friction:** When the vehicle stops, the work done by the frictional force ($f$) opposes the motion and dissipates the kinetic energy. The frictional force on a horizontal road is $f = \mu N = \mu mg$.
   The work done over stopping distance $s$ is $W = f \cdot s = \mu mg s$ [Source 64, 80].
3. **Work-Energy Theorem:** According to the theorem, the work done by the retarding force equals the change in kinetic energy (magnitude wise):
   $$ \mu mg s = K = \frac{p^2}{2m} $$
4. **Solving for Stopping Distance ($s$):**
   $$ s = \frac{p^2}{2m(\mu mg)} = \frac{p^2}{2 \mu m^2 g} $$

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