Question

A truck moving with a constant velocity \( 12 \, \text{ms}^{-1} \) crosses a car moving from rest with uniform acceleration \( 2 \, \text{ms}^{-2} \). The distance the car has to travel from the starting point to cross the truck again is:

• A. 50 m
• B. 60 m
• C. 144 m
• D. 120 m

Hint:

For problems involving relative motion, set up equations for distances traveled by both objects and solve for the time at which they meet.

Answer 1

The Correct Option is C

Let the time taken for the car to cross the truck be \( t \). - Distance traveled by the truck in time \( t \) is: \[ d_{\text{truck}} = 12t \] - Distance traveled by the car in time \( t \) with an initial velocity of 0 and acceleration \( 2 \, \text{ms}^{-2} \) is: \[ d_{\text{car}} = \frac{1}{2} \times 2t^2 = t^2 \] Since the car needs to cross the truck, the distance traveled by the car should be equal to the distance traveled by the truck plus the length of the truck. We assume the truck's length to be negligible. Thus, the equation becomes: \[ t^2 = 12t \] Solving for \( t \), we get: \[ t = 12 \, \text{seconds} \] Now, substituting \( t = 12 \) into the equation for the distance traveled by the car: \[ d_{\text{car}} = 12^2 = 144 \, \text{m} \] Thus, the distance the car has to travel to cross the truck again is \( 144 \, \text{m} \). The correct answer is option (3), \( 144 \, \text{m} \).

Answer 2

Step 1: Write down the given data
Velocity of truck, \( v = 12 \, \text{ms}^{-1} \) (constant)
Car starts from rest with acceleration, \( a = 2 \, \text{ms}^{-2} \)
Initial velocity of car, \( u = 0 \)

Step 2: Define variables and equations
Let the time taken by the car to catch the truck again be \( t \).
Distance travelled by truck in time \( t \): \( s_{truck} = v t = 12t \)
Distance travelled by car in time \( t \): \( s_{car} = ut + \frac{1}{2} a t^2 = 0 + \frac{1}{2} \times 2 \times t^2 = t^2 \)

Step 3: Set distances equal to find time when car crosses truck again
\[ s_{car} = s_{truck} \implies t^2 = 12t \]
\[ t^2 - 12t = 0 \implies t(t - 12) = 0 \]
So, \( t = 0 \) (initial time) or \( t = 12 \, \text{s} \)

Step 4: Calculate distance travelled by car at \( t = 12 \) s
\[ s_{car} = t^2 = 12^2 = 144 \, \text{m} \]

Final answer: The car travels 144 meters from the starting point to cross the truck again.