# Mechanics

calculus-based physics for scientists and engineers

## Getting the Position from the Acceleration

### Going from Acceleration to Position using Antiderivatives

It turns out that rarely in real life do we know the position function and need the acceleration. It is the other way around. If we use derivatives to go from position to acceleration, we can use anti-derivatives to go from acceleration to position! But, there is a catch ...

### Velocity Differences

You can calculate velocity differences with this one weird trick! Just click on the link! Do it!

### Position Differences

Let us do the same thing with position differences.

### Solution

1.

The figure below shows the velocity of a particle as a function of time. The particle starts at $$x(0) = 8$$ m. Does the particle have a turning point? If so, when? What is the position at 1, 2, and 4 seconds? Sketch graphs for both the position and acceleration function.

### Solution

2. For falling objects near the surface of the earth our model says the acceleration is a constant value, $$g$$. If we make the positive axis point up, the acceleration is $$a(t)=-g$$. Derive the velocity and position functions for arbitrary initial conditions, $$x_o$$ and $$v_o$$.