A car and a motorcycle are moving with the same momentum. When equal retarding forces are applied, the car comes to halt in `t_1` seconds and the motorcycle in `t_2` seconds. If the mass of the car is five times more than the mass of the motorcycle then

Options

A. `t_1=t_2`
B. `t_1=5t_2`
C. `t_1=1/5 t_2`
D. `t_1=25 t_2`

Explanation

To solve the problem step by step, we will analyze the given information and apply the principles of physics.

### Step 1: Understand the Given Information
- A car and a motorcycle have the same momentum.
- The mass of the car (m_c) is five times the mass of the motorcycle (m_m).
- Equal retarding forces are applied to both vehicles.
- The car comes to a halt in time \( t_1 \) and the motorcycle in time \( t_2 \).

### Step 2: Define the Momentum
Since both vehicles have the same momentum, we can express their momentum as:
- Momentum of the car, \( P_c = m_c \cdot v_c \)
- Momentum of the motorcycle, \( P_m = m_m \cdot v_m \)

Given that \( P_c = P_m \), we can write:
\[ m_c \cdot v_c = m_m \cdot v_m \]

### Step 3: Relate the Masses
We know that:
\[ m_c = 5 \cdot m_m \]
Substituting this into the momentum equation gives:
\[ 5 \cdot m_m \cdot v_c = m_m \cdot v_m \]
Dividing both sides by \( m_m \) (assuming \( m_m \neq 0 \)):
\[ 5 \cdot v_c = v_m \]
Thus:
\[ v_m = 5 \cdot v_c \]

### Step 4: Apply Newton's Second Law
According to Newton's second law, the force is equal to the rate of change of momentum:
- For the car:
\[ F = \frac{\Delta P_c}{\Delta t_1} = \frac{0 - P_c}{t_1} = -\frac{P_c}{t_1} \]
- For the motorcycle:
\[ F = \frac{\Delta P_m}{\Delta t_2} = \frac{0 - P_m}{t_2} = -\frac{P_m}{t_2} \]

Since the retarding forces are equal, we can set these two equations equal to each other:
\[ -\frac{P_c}{t_1} = -\frac{P_m}{t_2} \]
This simplifies to:
\[ \frac{P_c}{t_1} = \frac{P_m}{t_2} \]

### Step 5: Substitute the Momentum Values
Since \( P_c = P_m \), we can substitute:
\[ \frac{P}{t_1} = \frac{P}{t_2} \]
This leads to:
\[ \frac{1}{t_1} = \frac{1}{t_2} \]
Cross-multiplying gives:
\[ t_1 = t_2 \]

### Conclusion
Thus, we conclude that:
\[ t_1 = t_2 \]

### Final Answer
The correct relation is that the time taken for the car to come to a halt is equal to the time taken for the motorcycle to come to a halt, i.e., \( t_1 = t_2 \).

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