A car is driving a long a straight horizontal road as shown in figure 1. Starting from rest at time t = 0 the car moves with an acceleration a that varies with time t according to the equation a(t) = P - Qt where P and Q are positive constants and the positive direction is taken to be toward the right. Derive an equation for the car's velocity v as a function of time t

To derive an equation for the car's velocity \( v(t) \) as a function of time \( t \), we start by recognizing that acceleration is the time derivative of velocity. The relationship between acceleration and velocity is given by:

\[
a(t) = \frac{dv}{dt}
\]

Given the acceleration function \( a(t) = P - Qt \), we can integrate this with respect to \( t \) to find the velocity as a function of time.

### Step 1: Write down the differential equation
\[
\frac{dv}{dt} = P - Qt
\]

### Step 2: Integrate both sides with respect to time
To find \( v(t) \), we integrate the right-hand side with respect to \( t \):

\[
v(t) = \int (P - Qt) \, dt
\]

Perform the integration:

\[
v(t) = P t - \frac{Q t^2}{2} + C
\]

where \( C \) is the constant of integration.

### Step 3: Apply the initial condition
We are told that the car starts from rest at \( t = 0 \), which implies that the initial velocity is zero:

\[
v(0) = 0
\]

Substitute \( t = 0 \) into the equation for \( v(t) \):

\[
0 = P(0) - \frac{Q(0)^2}{2} + C \implies C = 0
\]

Thus, the constant \( C \) is zero.

### Step 4: Final velocity equation
The velocity as a function of time is:

\[
v(t) = P t - \frac{Q t^2}{2}
\]

This equation describes the car's velocity as a function of time given the time-varying acceleration.

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Do you have any questions or would you like further details? 

Here are 5 related questions:
1. How would the velocity change if \( P \) were larger?
2. What happens to the velocity when \( t = \frac{2P}{Q} \)?
3. How does this velocity function relate to the displacement of the car?
4. What would the acceleration look like graphically as a function of time?
5. How would the equation change if \( Q \) were negative?

**Tip**: When integrating, always remember to include the constant of integration and use initial conditions to determine its value.

A car is driving along a straight horizontal road as shown in figure 1. Starting from rest at time t = 0 the car moves with an acceleration a that varies with time t according to the equation a(t) = P - Qt where P and Q are positive constants and the positive direction is taken to be toward the right. Derive an equation for the car's velocity v as a function of time t.

Explore how to derive the velocity equation for a car moving with time-varying acceleration given by a(t) = P - Qt. This problem involves calculus, specifically integration, to find the velocity function v(t). Ideal for students in Grades 11-12 or college-level physics courses.

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