Question

Ram has just learnt driving and he is driving on a wet straight road ( μS = 0.1, μk = 0.05 ) with a speed of 108 km/hr. He sees his friend Shyam ahead travelling at a constant speed of 36 km/hr in the same direction. His horn fails, he is at a distance of 102 m from Shyam. He applies brakes just hard enough to prevent slipping, yet providing for maximum deceleration. If horn had worked and Shyam started accelerating

• A. Accident could have been avoided only if Shyam accelerated at maximum possible rate too.
• B. Accident could not have been avoided.
• C. Accident could have been avoided even if Shyam did not accelerate at maximum possible rate.
• D. Accident could have been avoided even if Shyam did not accelerate at all, but moved at same speed as before.

Answer 1

Answer: (C)

C

Answer 2

The problem involves relative motion and braking/acceleration. We need to determine if a collision can be avoided under different conditions.

1. Convert Speeds to m/s:

• Ram's initial speed, $V_R = 108 \text{ km/hr} = 108 \times \frac{5}{18} \text{ m/s} = 30 \text{ m/s}$.
• Shyam's initial speed, $V_S = 36 \text{ km/hr} = 36 \times \frac{5}{18} \text{ m/s} = 10 \text{ m/s}$.
• Initial distance between them, $D = 102 \text{ m}$.

2. Calculate Ram's Maximum Deceleration:

Ram applies brakes "just hard enough to prevent slipping, yet providing for maximum deceleration". This means the braking force is due to maximum static friction.

• Maximum static friction force, $f_{s,max} = \mu_S \times N = \mu_S \times mg$.
• By Newton's second law, $f_{s,max} = m \times a_R$, where $a_R$ is Ram's deceleration.
• So, $m \times a_R = \mu_S \times mg \implies a_R = \mu_S \times g$.
• Given $\mu_S = 0.1$ and assuming $g = 10 \text{ m/s}^2$, $a_R = 0.1 \times 10 = 1 \text{ m/s}^2$.

3. Analyze Collision Avoidance using Relative Motion:

Let's consider the motion of Ram relative to Shyam.

• Initial relative velocity, $u_{rel} = V_R - V_S = 30 - 10 = 20 \text{ m/s}$.
• Ram's acceleration (deceleration) is $-a_R = -1 \text{ m/s}^2$.
• Let Shyam's acceleration be $a_S$.
• Relative acceleration, $a_{rel} = a_R - a_S = -1 - a_S = -(1 + a_S)$. (This is the acceleration of Ram with respect to Shyam).

For collision avoidance, Ram's relative velocity with respect to Shyam must become zero (or negative, meaning Ram falls behind Shyam) before the relative distance becomes zero. Using the kinematic equation $v_{rel}^2 = u_{rel}^2 + 2 a_{rel} s_{rel}$, where $v_{rel} = 0$ (final relative velocity).

• $0^2 = (20)^2 + 2 \times (-(1 + a_S)) \times s_{rel}$
• $0 = 400 - 2(1 + a_S) s_{rel}$
• $s_{rel} = \frac{400}{2(1 + a_S)} = \frac{200}{1 + a_S}$

This $s_{rel}$ is the minimum distance Ram needs to cover relative to Shyam to make their speeds equal. For no collision, this distance must be less than or equal to the initial separation $D$.

• $s_{rel} \le D$
• $\frac{200}{1 + a_S} \le 102$
• $200 \le 102 (1 + a_S)$
• $1 + a_S \ge \frac{200}{102}$
• $1 + a_S \ge 1.96078...$
• $a_S \ge 0.96078...\text{ m/s}^2$

4. Evaluate Shyam's Maximum Possible Acceleration:

Shyam is also on the same wet road. His maximum possible acceleration (assuming engine power is sufficient to utilize maximum static friction without wheel spin) is also given by $\mu_S \times g$.

• $a_{S,max} = \mu_S \times g = 0.1 \times 10 = 1 \text{ m/s}^2$.

5. Analyze the Options:

• Option (D): Accident could have been avoided even if Shyam did not accelerate at all, but moved at same speed as before.

  If $a_S = 0$, then the required relative distance $s_{rel} = \frac{200}{1 + 0} = 200 \text{ m}$. Since the initial distance $D = 102 \text{ m}$ is less than $200 \text{ m}$, an accident would occur. So, option (D) is false.

• Option (B): Accident could not have been avoided.

  We found that if Shyam accelerates at $a_S \ge 0.96078 \text{ m/s}^2$, the accident can be avoided. Since Shyam's maximum possible acceleration is $1 \text{ m/s}^2$, which is greater than $0.96078 \text{ m/s}^2$, it is possible for Shyam to accelerate enough to avoid the accident. So, option (B) is false.

• Option (A): Accident could have been avoided only if Shyam accelerated at maximum possible rate too.

  The required acceleration for Shyam is $a_S \ge 0.96078 \text{ m/s}^2$. Shyam's maximum possible acceleration is $1 \text{ m/s}^2$. Since $0.96078 \text{ m/s}^2 < 1 \text{ m/s}^2$, Shyam does not need to accelerate at his maximum possible rate to avoid the accident. He can accelerate at a rate slightly above 0.96078 m/s² (e.g., 0.97 m/s²) which is less than his maximum rate. So, option (A) is false.

• Option (C): Accident could have been avoided even if Shyam did not accelerate at maximum possible rate.

  As established above, Shyam needs to accelerate at $a_S \ge 0.96078 \text{ m/s}^2$. His maximum possible acceleration is $1 \text{ m/s}^2$. Since the required acceleration is less than his maximum possible acceleration, he can avoid the accident by accelerating at a rate that is not his maximum. For example, if he accelerates at 0.97 m/s², the accident is avoided, and 0.97 m/s² is not his maximum rate of 1 m/s². So, option (C) is true.