A truck and a car moving with the same kinetic energy are brought to rest by the application of brakes which provide equal retarding forces. Which of them will come to rest in a shorter distance?

Options

A. The truck
B. The car
C. Both will travel the same distance before coming to rest
D. Cannot be predicted

Explanation

To solve the problem, we need to analyze the situation using the concepts of kinetic energy, work done by the retarding force, and the equations of motion.

### Step 1: Understand the given information
We have two vehicles: a truck and a car, both moving with the same kinetic energy (KE). They are brought to rest by applying brakes that provide equal retarding forces (F).

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

### Step 3: Set up the equations for both vehicles
Let:
- \( m_t \) = mass of the truck
- \( m_c \) = mass of the car
- \( v_t \) = velocity of the truck
- \( v_c \) = velocity of the car

Since both have the same kinetic energy:
\[ \frac{1}{2}m_t v_t^2 = \frac{1}{2}m_c v_c^2 \]
This implies:
\[ m_t v_t^2 = m_c v_c^2 \]

### Step 4: Use the work-energy principle
The work done by the retarding force (F) is equal to the change in kinetic energy. When the vehicles come to rest, their final kinetic energy is zero, so:
\[ W = -KE \]
The work done can also be expressed as:
\[ W = F \cdot d \]
where \( d \) is the distance over which the force acts.

### Step 5: Set up the equations for work done
For the truck:
\[ F \cdot d_t = m_t v_t^2 \]
For the car:
\[ F \cdot d_c = m_c v_c^2 \]

### Step 6: Solve for distances
From the equations above, we can express the distances:
\[ d_t = \frac{m_t v_t^2}{F} \]
\[ d_c = \frac{m_c v_c^2}{F} \]

### Step 7: Compare the distances
Since \( m_t v_t^2 = m_c v_c^2 \), we can substitute \( m_c v_c^2 \) into the equation for \( d_c \):
\[ d_t = \frac{m_t v_t^2}{F} \]
\[ d_c = \frac{m_c v_c^2}{F} = \frac{m_t v_t^2}{F} \]

### Step 8: Conclusion
Since both \( d_t \) and \( d_c \) are equal, we conclude that both the truck and the car will come to rest in the same distance when equal retarding forces are applied.

### Final Answer
Both the truck and the car will come to rest in the same distance.

---