If a car is driving in a straight line with constant acceleration a, initial velocity v0, and initial position s0, find a formula for the position function of the car.

How would I solve this equation?

dv/dt = a which is constant

so
v = c + at
c is v when t = 0, v0
so
v = v0 + a t

dx/dt = v = v0 + a t
so
x = constant + v0 t + (1/2)at^2
constant is x when t = 0 or s0
x = s0 + v0 t + (1/2)at^2

(but we cheated and took physics so knew that)

To find the position function of the car, which describes its position at any given time, we can use the equations of motion derived from the kinematic equations. Here's how you can solve this equation step-by-step:

1. Start with the equation for position as a function of time:
s(t) = s0 + v0t + (1/2)at^2

2. In this equation, s(t) represents the position of the car at time t, s0 represents the initial position of the car, v0 represents the initial velocity, a represents the constant acceleration, and t represents time.

3. Substitute the given values of s0, v0, and a into the equation.

4. Simplify the equation to get the final formula for the position function of the car.

For example, let's say the car starts at position s0 = 10 meters, with an initial velocity v0 = 2 m/s, and constant acceleration a = 3 m/s^2.

Plugging in these values, the position function of the car becomes:
s(t) = 10 + 2t + (1/2)(3)t^2

This equation will give you the position of the car at any given time t.