A particle starts from rest and has velocity kt after time t. The distance travelled in time t is-

Options

A. `kt^(2)`
B. `2kt^(2)`
C. `ksqrt(t)`
D. `1//2kt^(2)`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Understand the Given Information
- The particle starts from rest, which means its initial velocity \( u = 0 \).
- The velocity of the particle after time \( t \) is given by \( v = kt \).

### Step 2: Find the Acceleration
Using the first equation of motion:
\[ v = u + at \]
Substituting the known values:
\[ kt = 0 + at \]
This simplifies to:
\[ kt = at \]
Dividing both sides by \( t \) (assuming \( t \neq 0 \)):
\[ a = k \]
So, the acceleration \( a \) is \( k \).

### Step 3: Calculate the Distance Travelled
We will use the second equation of motion to find the distance \( s \):
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the values we have:
- \( u = 0 \)
- \( a = k \)

Thus, the equation becomes:
\[ s = 0 \cdot t + \frac{1}{2} k t^2 \]
This simplifies to:
\[ s = \frac{1}{2} k t^2 \]

### Conclusion
The distance travelled by the particle in time \( t \) is:
\[ s = \frac{1}{2} k t^2 \]