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Physics-

General

Easy

Question

Two cars $A$ and $B$ are travelling in the same direction with velocities $v subscript A end subscript$ and $v subscript B end subscript left parenthesis v subscript A end subscript greater than v subscript B end subscript right parenthesis$ . When the car $A$ is at a distance $s$ behind car $B$ , the driver of the car $A$ applies the brakes producing a uniform retardation $a$ , there will be no collision when

$s < \frac{{(v_{A} - v_{B})}^{2}}{2 a}$
$s = \frac{{(v_{A} - v_{B})}^{2}}{2 a}$
$s \geq \frac{{(v_{A} - v_{B})}^{2}}{2 a}$
$s \leq \frac{{(v_{A} - v_{B})}^{2}}{2 a}$

The correct answer is: $s greater or equal than fraction numerator open parentheses v subscript A end subscript minus v subscript B end subscript close parentheses to the power of 2 end exponent over denominator 2 a end fraction$

For no collision, the speed of car $A$ should be reduced to $v subscript B end subscript$ before the cars meet, $i e comma blank$ final relative velocity of car $A$ with respect to car $B$ is zero $i e$ , $v subscript r end subscript equals 0$
Here initial relative velocity, $u subscript r end subscript equals v subscript A end subscript minus v subscript B end subscript$
Relative acceleration, $a subscript r end subscript equals negative a minus 0 equals negative a$
Let relative displacement $equals s subscript r end subscript$
The equation
$v subscript r end subscript superscript 2 end superscript equals u subscript r end subscript superscript 2 end superscript plus 2 a subscript r end subscript s subscript r end subscript$
$open parentheses 0 close parentheses to the power of 2 end exponent equals open parentheses v subscript A end subscript minus v subscript B end subscript close parentheses minus 2 a s comma$
$s subscript r end subscript equals fraction numerator open parentheses v subscript A end subscript minus v subscript B end subscript close parentheses to the power of 2 end exponent over denominator 2 a end fraction$
For no collision, $s subscript r end subscript less or equal than s$
$i e comma blank fraction numerator open parentheses v subscript A end subscript minus v subscript B end subscript close parentheses to the power of blank over denominator 2 a end fraction less or equal than s$

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