A police van moving on a highway with a speed of $30 \mathrm{~km}_{h}^{-1}$ fires a bullet at a thief's car speeding away in a same direction with a speed of $192 \mathrm{~km} \mathrm{~h}^{-1}$. If the muzzle speed of the bullet $150 \mathrm{~m} \mathrm{~s}^{-1}$, with what speed does the bullet hit the thief's car?

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Aicharya · Accepted Answer

Speed of police van $=30 \times \frac{5}{18}=\frac{25}{3} \mathrm{~m} \mathrm{~s}^{-1}$ The muzzle velocity, that is, the velocity of bullet with respect to van is $\begin{aligned} \left[\vec{v}_{\text{{bullet}}}\right]_{\text{{van}}} &=\left[\vec{v}_{\text{{bullet}}}\right]_{\text{{ground}}} - \left[\vec{v}_{\text{{van}}}\right]_{\text{{ground}}} \ \left[\vec{v}_{\text{{bullet}}}\right]_{\text{{ground}}} &=\left[\vec{v}_{\text{{bullet}}}\right]_{\text{{van}}} + \left[\vec{v}_{\text{{van}}}\right]_{\text{{ground}}} \ &=150 + \frac{25}{3} = \frac{475}{3} \, \text{{m}} \, \text{{s}}^{-1} \end{aligned}$ Speed of thief's car $= 192 \times \frac{5}{18}$ = $\frac{160}{3} \, \text{{m}} \, \text{{s}}^{-1}$ Now velocity of bullet with respect to the thief's car $\begin{aligned} \left[\vec{v}_{\text{{bullet}}}\right]_{\text{{car}}} &=\left[\vec{v}_{\text{{bullet}}}\right]_{\text{{ground}}} - \left[\vec{v}_{\text{{car}}}\right]_{\text{{ground}}} \ &= \frac{475}{3} - \frac{160}{3} = 105 \, \text{{m}} \, \text{{s}}^{-1} \end{aligned}$ Hence, the bullet hits the thief's car with speed $105 \, \text{{m}} \, \text{{s}}^{-1}$.

A police van moving on a highway with a speed of \(30 \mathrm{~km}_{h}^{-1}\) fires a bullet at a thief's…

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