A truck and a car are moving on a smooth, level road such that the K.E. associated with them is same. Brakes are applied to both of them simultaneously. Which one will cover a greater distance before it stops?

Options

A. Car
B. Truck
C. Both will cover the same distance
D. Nothing can be decided

Answer

To solve the problem of which vehicle (the truck or the car) will cover a greater distance before stopping when both have the same kinetic energy, we can follow these steps:

### Step 1: Understand the Given Information
We know that both the truck and the car have the same kinetic energy. Let's denote:
- Mass of the truck = \( m_1 \)
- Mass of the car = \( m_2 \)
- Initial speed of the truck = \( v_1 \)
- Initial speed of the car = \( v_2 \)

The kinetic energy (K.E.) for both vehicles can be expressed as:
\[ K.E. = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
Since the kinetic energies are equal, we can write:
\[ m_1 v_1^2 = m_2 v_2^2 \]

### Step 2: Apply Newton's Laws of Motion
When brakes are applied, both vehicles experience the same retarding force \( F \). The acceleration (deceleration in this case) for each vehicle can be defined as:
- For the truck: \( a_1 = \frac{F}{m_1} \)
- For the car: \( a_2 = \frac{F}{m_2} \)

### Step 3: Use the Equation of Motion
We can use the equation of motion to find the stopping distance for each vehicle. The equation is:
\[ v^2 = u^2 + 2as \]
Where:
- \( v \) = final velocity (0 when the vehicle stops)
- \( u \) = initial velocity
- \( a \) = acceleration (negative in this case)
- \( s \) = distance traveled before stopping

For the truck:
\[ 0 = v_1^2 - 2a_1 s_1 \implies s_1 = \frac{v_1^2}{2a_1} = \frac{v_1^2}{2 \cdot \frac{F}{m_1}} = \frac{m_1 v_1^2}{2F} \]

For the car:
\[ 0 = v_2^2 - 2a_2 s_2 \implies s_2 = \frac{v_2^2}{2a_2} = \frac{v_2^2}{2 \cdot \frac{F}{m_2}} = \frac{m_2 v_2^2}{2F} \]

### Step 4: Compare the Distances
Now we have expressions for the stopping distances:
- Distance traveled by the truck: \( s_1 = \frac{m_1 v_1^2}{2F} \)
- Distance traveled by the car: \( s_2 = \frac{m_2 v_2^2}{2F} \)

Since we established earlier that \( m_1 v_1^2 = m_2 v_2^2 \), we can substitute this into our distance equations:
\[ s_1 = \frac{m_1 v_1^2}{2F} = \frac{m_2 v_2^2}{2F} = s_2 \]

### Conclusion
Thus, both the truck and the car will cover the same distance before stopping.

### Final Answer
**Both will cover the same distance before stopping.**

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