Question

A driver applies the brakes on seeing the red traffic signal $400\text{ m}$ ahead. At the time of applying the brakes, the vehicle was moving with a speed of $54\text{ km/h}$ and retards uniformly at $0.3\text{ m/s}^2$. The distance of the vehicle from the traffic signal when it stops is

• A. $25\text{ m}$
• B. $375\text{ m}$
• C. $350\text{ m}$
• D. $50\text{ m}$

Hint:

Be sure to read the final question carefully! Calculating a stopping distance of $375\text{ m}$ might tempt you to select option (B) immediately. Always double-check whether the question is asking for the distance traveled ($375\text{ m}$) or the remaining distance from the target object ($400 - 375 = 25\text{ m}$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A vehicle moving at an initial velocity sees a red light located $d_{\text{total}} = 400\text{ m}$ ahead and brakes with a constant deceleration of $a = -0.3\text{ m/s}^2$. We need to compute the final remaining distance between the car and the traffic signal once the vehicle comes to a complete stop ($v = 0$).

Step 2: Key Formula or Approach:
1. First, convert the initial speed from km/h to standard SI units (m/s) by multiplying by $\frac{5}{18}$. 2. Use Newton's third equation of motion to calculate the total braking distance $S$ required to stop the car: $$v^2 = u^2 + 2aS$$ 3. The remaining distance to the signal is found by subtracting this stopping distance from the initial total distance: $d_{\text{remaining}} = 400 - S$.

Step 3: Detailed Explanation:
Convert the initial velocity $u$ to m/s: $$u = 54 \times \frac{5}{18} = 3 \times 5 = 15\text{ m/s}$$ Given values: final velocity $v = 0$, and uniform acceleration $a = -0.3\text{ m/s}^2$ (negative due to retardation). Substitute these into the kinematic equation: $$0^2 = (15)^2 + 2(-0.3)S$$ $$0 = 225 - 0.6S$$ $$0.6S = 225 \implies S = \frac{225}{0.6} = \frac{2250}{6}$$ Divide by 6 to find the braking distance: $$S = 375\text{ meters}$$ Now, calculate the final clearance distance from the traffic signal: $$d_{\text{remaining}} = 400\text{ m} - 375\text{ m} = 25\text{ meters}$$

Step 4: Final Answer:
The distance of the vehicle from the traffic signal when it stops is $25\text{ m}$, which corresponds to option (A).