A 400kg car attains a speed of 50m/s from rest in 20s. The power developed in the engine will be

Options

A. 50W
B. 0.5W
C. 5kW
D. 25kW

Explanation

To solve the problem of finding the power developed in the engine of a car that accelerates from rest to a speed of 50 m/s in 20 seconds, we can follow these steps:

### Step 1: Calculate the Change in Kinetic Energy
The formula for kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} m v^2 \]
Where:
- \( m \) is the mass of the car (400 kg)
- \( v \) is the final velocity (50 m/s)

First, we calculate the initial kinetic energy (when the car is at rest, \( v = 0 \)):
\[ KE_{\text{initial}} = \frac{1}{2} \times 400 \times 0^2 = 0 \, \text{J} \]

Next, we calculate the final kinetic energy:
\[ KE_{\text{final}} = \frac{1}{2} \times 400 \times (50)^2 = \frac{1}{2} \times 400 \times 2500 = 500000 \, \text{J} \]

Now, we find the change in kinetic energy:
\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = 500000 \, \text{J} - 0 \, \text{J} = 500000 \, \text{J} \]

### Step 2: Calculate the Power Developed
Power is defined as the work done per unit time. In this case, the work done is equal to the change in kinetic energy:
\[ P = \frac{\text{Work}}{\text{Time}} = \frac{\Delta KE}{t} \]
Where:
- \( t \) is the time taken (20 s)

Substituting the values:
\[ P = \frac{500000 \, \text{J}}{20 \, \text{s}} = 25000 \, \text{W} \]

### Step 3: Convert Watts to Kilowatts
To convert watts to kilowatts, we divide by 1000:
\[ P = \frac{25000 \, \text{W}}{1000} = 25 \, \text{kW} \]

### Final Answer
The power developed in the engine is **25 kW**.

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