A constant power P is applied on a particle of mass m. find kintic energy, velocity and displacement of particle as function of time t.

Explanation

To solve the problem of finding the kinetic energy, velocity, and displacement of a particle of mass \( m \) under the influence of a constant power \( P \), we can follow these steps:

### Step 1: Relate Power to Work Done
We know that power \( P \) is defined as the rate of doing work, which can be expressed mathematically as:
\[ P = \frac{W}{t} \]
where \( W \) is the work done and \( t \) is the time. Rearranging this gives us:
\[ W = P \cdot t \]

### Step 2: Relate Work Done to Change in Kinetic Energy
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Thus:
\[ W = \Delta KE = KE - KE_0 \]
Assuming the particle starts from rest, \( KE_0 = 0 \), we have:
\[ \Delta KE = KE = W \]
Substituting the expression for work done, we get:
\[ KE = P \cdot t \]

### Step 3: Express Kinetic Energy in Terms of Velocity
The kinetic energy of a particle is given by:
\[ KE = \frac{1}{2} m v^2 \]
Setting this equal to the expression we found for kinetic energy:
\[ \frac{1}{2} m v^2 = P \cdot t \]

### Step 4: Solve for Velocity
Now, we can solve for the velocity \( v \):
\[ m v^2 = 2 P t \]
\[ v^2 = \frac{2 P t}{m} \]
Taking the square root gives:
\[ v = \sqrt{\frac{2 P t}{m}} \]

### Step 5: Express Velocity as a Function of Time
We can rewrite the expression for velocity:
\[ v = \sqrt{\frac{2P}{m}} \cdot t^{1/2} \]

### Step 6: Find Displacement as a Function of Time
To find displacement \( s \), we use the relationship:
\[ v = \frac{ds}{dt} \]
This implies:
\[ ds = v \cdot dt \]
Substituting the expression for \( v \):
\[ ds = \sqrt{\frac{2P}{m}} \cdot t^{1/2} \cdot dt \]
Integrating both sides gives:
\[ s = \int \sqrt{\frac{2P}{m}} \cdot t^{1/2} \, dt \]
The integral of \( t^{1/2} \) is:
\[ \int t^{1/2} \, dt = \frac{2}{3} t^{3/2} \]
Thus, we have:
\[ s = \sqrt{\frac{2P}{m}} \cdot \frac{2}{3} t^{3/2} + C \]
Assuming the initial displacement is zero (i.e., \( C = 0 \)):
\[ s = \frac{2}{3} \sqrt{\frac{2P}{m}} t^{3/2} \]

### Final Results
1. **Kinetic Energy**:
\[ KE = P \cdot t \]
2. **Velocity**:
\[ v = \sqrt{\frac{2P}{m}} \cdot t^{1/2} \]
3. **Displacement**:
\[ s = \frac{2}{3} \sqrt{\frac{2P}{m}} t^{3/2} \]