If a car has an acceleration function of \( a(t) = 6t \), what is its velocity function assuming it starts from rest?

To find the velocity function \( v(t) \) of a car given its acceleration function \( a(t) = 6t \) and the initial condition that the car starts from rest (i.e., \( v(0) = 0 \)), follow these steps: ### 1. Understand the Relationship Between Acceleration and Velocity Acceleration is the derivative of velocity with respect to time: \[ a(t) = \frac{dv(t)}{dt} \] To find the velocity function \( v(t) \), you need to integrate the acceleration function with respect to time. ### 2. Integrate the Acceleration Function Given: \[ a(t) = 6t \] Integrate \( a(t) \) to find \( v(t) \): \[ v(t) = \int a(t) \, dt = \int 6t \, dt \] \[ v(t) = 6 \int t \, dt \] \[ v(t) = 6 \left( \frac{t^2}{2} \right) + C \] \[ v(t) = 3t^2 + C \] where \( C \) is the constant of integration. ### 3. Apply the Initial Condition The car starts from rest, which means at time \( t = 0 \), the velocity \( v(0) = 0 \): \[ v(0) = 3(0)^2 + C = 0 \] \[ 0 + C = 0 \Rightarrow C = 0 \] ### 4. Write the Final Velocity Function Substitute \( C = 0 \) back into the equation: \[ v(t) = 3t^2 \] ### Conclusion The velocity function of the car is: \[ v(t) = 3t^2 \] This means that the velocity increases quadratically with time.