Power supplied to a particle of 1 kg varies with times as `P = t^(2)/2` watt. At t = 0, v= 0, the velocity of the particle at time t = 3s is

Options

A. 3 m/s
B. 4 m/s
C. 2 m/s
D. 1 m/s

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the relationship between power and work
Power (P) is defined as the rate of doing work, which can be expressed mathematically as:
\[ P = \frac{dW}{dt} \]
where \( W \) is the work done.

### Step 2: Express work done in terms of power
From the definition of power, we can express the differential work done \( dW \) as:
\[ dW = P \, dt \]
Substituting the given power function \( P = \frac{t^2}{2} \):
\[ dW = \frac{t^2}{2} \, dt \]

### Step 3: Integrate to find the total work done
To find the total work done from \( t = 0 \) to \( t = 3 \) seconds, we integrate:
\[ W = \int_{0}^{3} \frac{t^2}{2} \, dt \]
Calculating the integral:
\[ W = \frac{1}{2} \int_{0}^{3} t^2 \, dt = \frac{1}{2} \left[ \frac{t^3}{3} \right]_{0}^{3} = \frac{1}{2} \left( \frac{3^3}{3} - \frac{0^3}{3} \right) \]
\[ = \frac{1}{2} \left( \frac{27}{3} \right) = \frac{1}{2} \times 9 = \frac{9}{2} \, \text{J} \]

### Step 4: Use the work-energy theorem
According to the work-energy theorem, the work done on the particle is equal to the change in kinetic energy:
\[ W = \Delta KE = KE_{final} - KE_{initial} \]
Given that the initial velocity \( v_0 = 0 \) at \( t = 0 \), the initial kinetic energy \( KE_{initial} = 0 \). Therefore:
\[ W = KE_{final} = \frac{1}{2} mv^2 \]
where \( m = 1 \, \text{kg} \) (mass of the particle) and \( v \) is the final velocity at \( t = 3 \, \text{s} \).

### Step 5: Set up the equation and solve for velocity
Substituting the work done:
\[ \frac{9}{2} = \frac{1}{2} \times 1 \times v^2 \]
This simplifies to:
\[ \frac{9}{2} = \frac{1}{2} v^2 \]
Multiplying both sides by 2:
\[ 9 = v^2 \]
Taking the square root of both sides:
\[ v = 3 \, \text{m/s} \]

### Final Answer
The velocity of the particle at \( t = 3 \, \text{s} \) is \( 3 \, \text{m/s} \).

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