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

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UpStudy AI · Accepted Answer

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.
The velocity function is $$ v(t) = 3t^2 $$.

If a car has an acceleration function of , what is its velocity function assuming it starts from rest?

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