Two cars of unequal masses use similar tyres. If they are moving at the same initial speed, the minimum stopping distance

Options

A. is smaller for the heavier
B. is smaller for the lighter car
C. is same for both cars
D. depends on the volume of the car

Answer

To solve the problem of determining the minimum stopping distance for two cars of unequal masses moving at the same initial speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Car**:
- When a car is moving and the brakes are applied, the frictional force acts in the opposite direction of motion. This frictional force is what causes the car to decelerate.

2. **Identify the Frictional Force**:
- The frictional force \( F_f \) can be expressed as:
\[ F_f = \mu N \]
- Where \( \mu \) is the coefficient of friction and \( N \) is the normal reaction force. For a car on a flat surface, \( N = mg \) (where \( m \) is the mass of the car and \( g \) is the acceleration due to gravity).

3. **Express the Frictional Force in Terms of Mass**:
- Substituting \( N \) into the frictional force equation gives:
\[ F_f = \mu mg \]

4. **Applying Newton's Second Law**:
- According to Newton's second law, the net force acting on the car is equal to the mass of the car times its acceleration \( a \):
\[ -F_f = ma \]
- Substituting the expression for \( F_f \):
\[ -\mu mg = ma \]

5. **Simplifying the Equation**:
- Dividing both sides by \( m \) (mass of the car):
\[ -\mu g = a \]
- This shows that the acceleration (deceleration in this case) is independent of the mass of the car.

6. **Using the Kinematic Equation to Find Stopping Distance**:
- We can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
- Where \( v \) is the final velocity (0 when the car stops), \( u \) is the initial velocity, \( a \) is the acceleration, and \( s \) is the stopping distance.
- Rearranging gives:
\[ 0 = u^2 + 2(-\mu g)s \]
- This simplifies to:
\[ s = \frac{u^2}{2\mu g} \]

7. **Conclusion**:
- The stopping distance \( s \) is given by the formula \( s = \frac{u^2}{2\mu g} \).
- Notably, this equation does not contain the mass \( m \) of the car, indicating that the stopping distance is independent of the mass of the car.
- Since both cars have the same initial speed \( u \) and use similar tires (same \( \mu \)), the stopping distance will be the same for both cars.

### Final Answer:
The minimum stopping distance is the same for both cars (Option 3).

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