A sports car accelerations from zero to a certain speed in `t` seconds. How long does it take for it to accelerate to twice that speed starting from rest, assuming the power of the engine to be constant (independent of velocity) and neglecting any resistance to motion?

Options

A. `sqrt(2)tsec`
B. `2tsec`
C. `3tsec`
D. `4tsec`

Explanation

To solve the problem, we need to analyze the relationship between power, time, and speed for the sports car under constant power conditions.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The sports car accelerates from rest (initial speed = 0) to a speed \( v \) in time \( t \).
- We need to find the time \( t' \) it takes to accelerate to a speed of \( 2v \) starting from rest.

2. **Using the Power Formula**:
- The power \( P \) is defined as the rate of doing work. The work done on the car to accelerate it to speed \( v \) is given by the kinetic energy formula:
\[ \text{Work} = \frac{1}{2} m v^2 \]
- The power can also be expressed in terms of time:
\[ P = \frac{\text{Work}}{t} = \frac{\frac{1}{2} m v^2}{t} \]
- Rearranging gives us:
\[ P \cdot t = \frac{1}{2} m v^2 \quad \text{(1)} \]

3. **Calculating Work for Speed \( 2v \)**:
- Now, we need to find the work done to accelerate the car to speed \( 2v \):
\[ \text{Work} = \frac{1}{2} m (2v)^2 = \frac{1}{2} m \cdot 4v^2 = 2 m v^2 \]
- The power in terms of the new time \( t' \) is:
\[ P \cdot t' = 2 m v^2 \quad \text{(2)} \]

4. **Relating the Two Equations**:
- From equation (1), we have:
\[ P \cdot t = \frac{1}{2} m v^2 \]
- From equation (2), we can express \( 2 m v^2 \) in terms of \( P \cdot t \):
\[ P \cdot t' = 2 m v^2 = 4 \left(\frac{1}{2} m v^2\right) = 4 P \cdot t \]
- Thus, we can equate:
\[ P \cdot t' = 4 P \cdot t \]

5. **Solving for \( t' \)**:
- Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ t' = 4t \]

6. **Conclusion**:
- Therefore, the time taken to accelerate to twice the speed \( 2v \) starting from rest is:
\[ t' = 4t \]

### Final Answer:
The time taken to accelerate to twice the speed is \( 4t \).