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If a $100 \mathrm{~m}$ long train needs $7.2 \mathrm{~s}$ to cross an object moving in a direction opposite to the train's direction with a speed of $5 \mathrm{~km} / \mathrm{h}$, then find the velocity of the train.
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Verified Answer
The correct answer is:
$45 \mathrm{~km} / \mathrm{h}$
Given, length of train, $l=100 \mathrm{~m}$
Time taken by an object to cross the train, $t=7.2 \mathrm{~s}$
Velocity of object opposite to train, $v_0=5 \mathrm{~km} \mathrm{~h}^{-1}$
$\begin{aligned} & =\frac{25}{18} \mathrm{~m} / \mathrm{s} \\ & =1.39 \mathrm{~m} / \mathrm{s}\end{aligned}$
Let the velocity of train, $v_t=v$
Using the concept of relative motion.
Relative velocity of train w.r.t object $v_{\text {rel }}=v_t+v_0$
Now, $l=v_{\text {rel }} \times t$
$100=7.2\left(v_t+v_0\right)$
$\Rightarrow \quad 100=7.2(v+1.39)$
$\Rightarrow \quad v=12.5 \mathrm{~m} / \mathrm{s}=45 \mathrm{~km} \mathrm{~h}^{-1}$
Time taken by an object to cross the train, $t=7.2 \mathrm{~s}$
Velocity of object opposite to train, $v_0=5 \mathrm{~km} \mathrm{~h}^{-1}$
$\begin{aligned} & =\frac{25}{18} \mathrm{~m} / \mathrm{s} \\ & =1.39 \mathrm{~m} / \mathrm{s}\end{aligned}$
Let the velocity of train, $v_t=v$
Using the concept of relative motion.
Relative velocity of train w.r.t object $v_{\text {rel }}=v_t+v_0$
Now, $l=v_{\text {rel }} \times t$
$100=7.2\left(v_t+v_0\right)$
$\Rightarrow \quad 100=7.2(v+1.39)$
$\Rightarrow \quad v=12.5 \mathrm{~m} / \mathrm{s}=45 \mathrm{~km} \mathrm{~h}^{-1}$
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