Velocity is the integral of the acceleration function, and distance is the integral of the velocity function. This is how we derive our equations of motion like d(t) = d0 + vt + (1/2)at2. These equations of motion assume acceleration is constant, acceleration = a. What if it varied over time --- like a multi-stage rocket?
If the acceleration is given by acceleration = at, I'm curious what kind of equations of motion you would get. Using indefinite integrals, develop general expressions for the velocity function, v(t), and for the distance function, d(t).

Great question! If the acceleration varies over time, it introduces a more complex system compared to the case where acceleration is constant. You've given a form for acceleration: \( a(t) = a_0 t \), where \( a_0 \) is a constant.

### Step 1: Define the problem
We know:
- \( a(t) = a_0 t \) (acceleration as a function of time),
- The velocity function \( v(t) \) is the integral of the acceleration \( a(t) \),
- The position function \( d(t) \) is the integral of the velocity function \( v(t) \).

Let's proceed with the integrations.

### Step 2: Find the velocity function \( v(t) \)

Velocity is the integral of acceleration, so:
\[
v(t) = \int a(t) \, dt = \int a_0 t \, dt
\]

Using basic integration rules:
\[
v(t) = a_0 \int t \, dt = a_0 \cdot \frac{t^2}{2} + C_1
\]

Where \( C_1 \) is the constant of integration, which can be interpreted as the initial velocity \( v_0 \) (the velocity at \( t = 0 \)).

Thus, the velocity function becomes:
\[
v(t) = \frac{a_0 t^2}{2} + v_0
\]

### Step 3: Find the position function \( d(t) \)

Now, the position function \( d(t) \) is the integral of the velocity function \( v(t) \). So:
\[
d(t) = \int v(t) \, dt = \int \left( \frac{a_0 t^2}{2} + v_0 \right) dt
\]

We integrate term by term:
\[
d(t) = \frac{a_0}{2} \int t^2 \, dt + v_0 \int dt
\]
\[
d(t) = \frac{a_0}{2} \cdot \frac{t^3}{3} + v_0 t + C_2
\]

Where \( C_2 \) is another constant of integration, which can be interpreted as the initial position \( d_0 \) (the position at \( t = 0 \)).

Thus, the position function becomes:
\[
d(t) = \frac{a_0 t^3}{6} + v_0 t + d_0
\]

### Final Equations of Motion

Given that the acceleration \( a(t) = a_0 t \), the resulting equations of motion are:

1. **Velocity function**:
   \[
   v(t) = \frac{a_0 t^2}{2} + v_0
   \]

2. **Position function**:
   \[
   d(t) = \frac{a_0 t^3}{6} + v_0 t + d_0
   \]

### Interpretation

- The velocity increases with the square of time, because the acceleration is increasing linearly with time.
- The position increases with the cube of time, reflecting the accelerated rate of velocity change.

### Do you want further clarification on these equations, or is there any part you'd like me to explain in more detail?

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### Further Questions:
1. How do these equations compare with the standard ones used when acceleration is constant?
2. How would the equations change if acceleration was proportional to \( t^2 \) instead of \( t \)?
3. What happens to the motion if you set \( v_0 = 0 \) and \( d_0 = 0 \)?
4. How would you derive equations of motion if acceleration were given as a sinusoidal function, like \( a(t) = a_0 \sin(\omega t) \)?
5. How would you interpret these equations in terms of real-world physical systems, such as a rocket with changing thrust?

### Tip:
When integrating, always remember to account for initial conditions (such as \( v_0 \) and \( d_0 \)) as constants of integration, as they play a crucial role in the physical interpretation of the solution.

Velocity is the integral of the acceleration function, and distance is the integral of the velocity function. This is how we derive our equations of motion like d(t) = d0 + vt + (1/2)at2. These equations of motion assume acceleration is constant, acceleration = a. What if it varied over time --- like a multi-stage rocket? If the acceleration is given by acceleration = at, I'm curious what kind of equations of motion you would get. Using indefinite integrals, develop general expressions for the velocity function, v(t), and for the distance function, d(t).

Explore how to derive the velocity and position functions for a system where acceleration varies with time, such as a multi-stage rocket. This solution uses indefinite integrals to develop expressions for v(t) and d(t) when acceleration is given as a(t) = at. Suitable for students in Grades 11-12 studying calculus and physics.

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