A car of mass m has an engine which can deliver constant power P.the maximum speed that the car can attain in t seconds in

Options

A. `sqrt((3)/(4)(Pt)/(m))`
B. `2sqrt((Pt)/(m))`
C. `sqrt((2Pt)/(m))`
D. `sqrt((Pt)/(m))`

Answer

To solve the problem of finding the maximum speed that a car can attain in \( t \) seconds given that it has a mass \( m \) and its engine delivers constant power \( P \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Work, Power, and Time**:
- Power is defined as the rate at which work is done. The relationship can be expressed as:
\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \implies \text{Work} = P \cdot t \]

2. **Apply the Work-Energy Theorem**:
- According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. For the car starting from rest, the initial kinetic energy is zero. Therefore, the work done on the car can be expressed as:
\[ \text{Work} = \Delta KE = \frac{1}{2} m v^2 - 0 = \frac{1}{2} m v^2 \]

3. **Set the Work Done Equal to the Change in Kinetic Energy**:
- From the previous steps, we can equate the work done to the change in kinetic energy:
\[ P \cdot t = \frac{1}{2} m v^2 \]

4. **Solve for Maximum Speed \( v \)**:
- Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{2Pt}{m} \]
- Taking the square root of both sides gives us the maximum speed \( v \):
\[ v = \sqrt{\frac{2Pt}{m}} \]

5. **Final Expression for Maximum Speed**:
- Thus, the maximum speed that the car can attain in \( t \) seconds is:
\[ v = \sqrt{\frac{2Pt}{m}} \]

### Conclusion:
The maximum speed that the car can attain in \( t \) seconds is given by:
\[ v = \sqrt{\frac{2Pt}{m}} \]