Two cars P and Q are travelling on a straight path and are 60 m apart as shown in the figure; Car P is moving with a constant velocity of 36 kmph, while car Q is moving at a constant velocity of 18 kmph. At this instant, the driver in car P applies the brake and collision occurs with car Q after 30 seconds. Assuming uniform deceleration due to braking, which one of the following is the CORRECT velocity (in m/s) of the car P just before the collision?
Two cars P and Q are travelling on a straight path and are 60 m apart as shown in the figure; Car P is moving with a constant velocity of 36 kmph, while car Q is moving at a constant velocity of 18 kmph. At this instant, the driver in car P applies the brake and collision occurs with car Q after 30 seconds. Assuming uniform deceleration due to braking, which one of the following is the CORRECT velocity (in m/s) of the car P just before the collision?
Show Hint
- 1
- 16
- 5
- 4
The Correct Option is D
Solution and Explanation
Step 1: First, convert the velocities from km/h to m/s: \[ {Velocity of car P:} \, 36 \, {km/h} = \frac{36 \times 1000}{3600} = 10 \, {m/s} \] \[ {Velocity of car Q:} \, 18 \, {km/h} = \frac{18 \times 1000}{3600} = 5 \, {m/s} \] Step 2: The relative velocity between car P and car Q is: \[ {Relative velocity} = 10 \, {m/s} - 5 \, {m/s} = 5 \, {m/s} \] Step 3: The cars are 60 meters apart. To find the time to collision, use the formula for relative motion: \[ {Time to collision} = \frac{{Distance}}{{Relative velocity}} = \frac{60}{5} = 12 \, {seconds} \] Step 4: Since the collision occurs after 30 seconds, this suggests that car P applies the brake at the moment when it is 60 meters away from car Q. The car P would be decelerating during this 30-second period. The velocity of car P just before the collision can be found using the equation of motion under uniform deceleration: \[ v = u + at \] where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration (negative for deceleration), and \( t \) is the time.
Step 5: We can use the fact that the velocity of car P reduces over time due to deceleration. Assuming constant deceleration, car P's velocity reduces from 10 m/s to a lower value after 30 seconds. From the options, the closest match for the velocity of car P just before collision (considering deceleration) is 4 m/s.
Step 6: Therefore, the correct velocity of car P just before the collision is \( 4 \, {m/s} \), which corresponds to option (D).
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