A

Step 1: Given the acceleration is a = kt.
Step 2: We know that acceleration is the derivative of velocity with respect to time, so we can write: \( a = \frac{dv}{dt} = kt \).
Step 3: To find velocity, we rearrange this equation: \( dv = kt \, dt \).
Step 4: Integrating both sides, we get: \( \int dv = \int kt \, dt\).
This leads to: \( v = \frac{kt^2}{2} + C \) where C is the constant of integration.
Step 5: Since the motion starts from rest, we have \( v(0) = 0 \), which gives us C = 0.
Step 6: Therefore, the velocity as a function of time is: \( v = \frac{kt^2}{2} \).
Step 7: Hence, we find the velocity in terms of t as: \( v = \frac{k}{2} t^2 \).
Thus, the final answer is \, \( \frac{k}{2} \).