Identical constant forces push two identical cars A and B continuously from a starting line to a finish line. The cars move on a frictionless horizontal surface. If car A is initially at rest and car-B is initially moving right with speed `v_(0)`. Choose the correct statement

Options

A. (a) Car-A has the larger change in momentum.
B. (b)Car-B has the larger change in momentum
C. (c)Both cars have the same change in momentum
D. (d)Not enough information is given decide.

Explanation

To solve the problem, we need to analyze the motion of the two identical cars A and B under the influence of identical constant forces. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Conditions
- Car A is initially at rest, so its initial velocity \( v_{A0} = 0 \).
- Car B is initially moving to the right with speed \( v_0 \), so its initial velocity \( v_{B0} = v_0 \).

### Step 2: Define the Forces and Acceleration
- Both cars are subjected to the same constant force \( F \).
- Since the cars are identical, they have the same mass \( m \).
- The acceleration \( a \) of both cars can be expressed as:
\[ a = \frac{F}{m} \]

### Step 3: Use Kinematic Equations
- For Car A (initially at rest):
- Final velocity \( v_A \) after traveling distance \( S \):
\[ v_A^2 = v_{A0}^2 + 2aS = 0 + 2aS = 2aS \]
- Therefore, \( v_A = \sqrt{2aS} \).

- For Car B (initially moving):
- Final velocity \( v_B \) after traveling the same distance \( S \):
\[ v_B^2 = v_{B0}^2 + 2aS = v_0^2 + 2aS \]
- Therefore, \( v_B = \sqrt{v_0^2 + 2aS} \).

### Step 4: Calculate Change in Momentum
- Change in momentum for Car A:
\[ \Delta p_A = m v_A = m \sqrt{2aS} \]

- Change in momentum for Car B:
\[ \Delta p_B = m v_B - m v_{B0} = m \left(\sqrt{v_0^2 + 2aS} - v_0\right) \]

### Step 5: Compare Changes in Momentum
To compare \( \Delta p_A \) and \( \Delta p_B \), we need to analyze the expressions:
- We know that both cars travel the same distance \( S \) and have the same acceleration \( a \).
- We can see that \( v_B \) will always be greater than \( v_A \) because Car B starts with an initial speed \( v_0 \).

### Step 6: Conclusion
- Since both cars are subjected to the same force and travel the same distance, but Car B starts with a higher initial velocity, the change in momentum of Car A will be greater than that of Car B.
- Therefore, the correct statement is:
\[ \Delta p_A > \Delta p_B \]

### Final Answer
The correct statement is that Car A has a larger change in momentum than Car B.