A truck and a car are moving on a smooth, level road such that the kinetic energy associated with them is same. Brakes are applied simultaneously in both of them such that equal retarding forces are produced in them. 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, we can follow these steps:

### Step 1: Understand the Given Information
We know that the truck and the car have the same kinetic energy when they are moving. Additionally, they experience equal retarding forces when brakes are applied.

### Step 2: Write the Kinetic Energy Equation
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \(m\) is the mass and \(v\) is the velocity of the object.

### Step 3: Set Up the Equations for Both Vehicles
Let:
- \(m_1\) = mass of the truck
- \(v_1\) = velocity of the truck
- \(m_2\) = mass of the car
- \(v_2\) = velocity of the car

Since the kinetic energies are equal, we can write:
\[ \frac{1}{2}m_1v_1^2 = \frac{1}{2}m_2v_2^2 \]
This simplifies to:
\[ m_1v_1^2 = m_2v_2^2 \]

### Step 4: Relate the Retarding Forces
The retarding force \(F\) acting on both vehicles is the same. According to Newton’s second law:
\[ F = ma \]
Thus, the acceleration (deceleration in this case) for each vehicle can be expressed as:
\[ a_1 = \frac{F}{m_1} \quad \text{(for the truck)} \]
\[ a_2 = \frac{F}{m_2} \quad \text{(for the car)} \]

### Step 5: Use the Equation of Motion
Using the equation of motion:
\[ v^2 = u^2 + 2as \]
where \(v\) is the final velocity (0 when stopped), \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the distance covered.

Rearranging gives:
\[ s = \frac{v^2}{2a} \]

### Step 6: Calculate Distances for Both Vehicles
For the truck:
\[ s_1 = \frac{v_1^2}{2a_1} = \frac{v_1^2}{2 \cdot \frac{F}{m_1}} = \frac{m_1v_1^2}{2F} \]

For the car:
\[ s_2 = \frac{v_2^2}{2a_2} = \frac{v_2^2}{2 \cdot \frac{F}{m_2}} = \frac{m_2v_2^2}{2F} \]

### Step 7: Substitute Kinetic Energy Relationship
Since we have established that \(m_1v_1^2 = m_2v_2^2\), we can substitute this into the expressions for \(s_1\) and \(s_2\):
\[ s_1 = \frac{m_1v_1^2}{2F} \quad \text{and} \quad s_2 = \frac{m_2v_2^2}{2F} \]
Since \(m_1v_1^2 = m_2v_2^2\), it follows that:
\[ s_1 = s_2 \]

### Conclusion
Both the truck and the car will cover the same distance before coming to a stop.