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Question: Two cars A and B are moving in the same direction with velocities 30 m/s and 20 m/s. When car A is a...

Two cars A and B are moving in the same direction with velocities 30 m/s and 20 m/s. When car A is at a distance dd behind car B, the driver of car A applies the brake producing a uniform retardation of 2 m/s². There will be no collision when:

A

d<25md < 25m

B

d>25md > 25m

C

d=25md = 25 m

D

d<12.5md < 12.5 m

Answer

d>25md > 25m

Explanation

Solution

The initial relative velocity of car A with respect to car B is urel=vA−vB=30 m/s−20 m/s=10 m/su_{rel} = v_A - v_B = 30 \text{ m/s} - 20 \text{ m/s} = 10 \text{ m/s}. The relative acceleration is arel=−2 m/s2a_{rel} = -2 \text{ m/s}^2. Using the kinematic equation vrel2=urel2+2arelsrelv_{rel}^2 = u_{rel}^2 + 2a_{rel}s_{rel}, where vrel=0v_{rel} = 0 (for stopping), we get: 02=(10 m/s)2+2(−2 m/s2)srel0^2 = (10 \text{ m/s})^2 + 2(-2 \text{ m/s}^2)s_{rel} 0=100 m2/s2−4 m/s2⋅srel0 = 100 \text{ m}^2/\text{s}^2 - 4 \text{ m/s}^2 \cdot s_{rel} srel=100 m2/s24 m/s2=25 ms_{rel} = \frac{100 \text{ m}^2/\text{s}^2}{4 \text{ m/s}^2} = 25 \text{ m}. For no collision, the initial distance dd must be greater than the relative stopping distance srels_{rel}. Therefore, d>25 md > 25 \text{ m}.