# Mechanics

calculus-based physics for scientists and engineers

## Motion with Constant Acceleration

### The Velocity and Position Functions for Constant Acceleration

If we let the acceleration be some non-zero constant, $$a$$ along the $$x$$ axis the functions for acceleration, velocity, and position are $$a(t) = a$$ $$v_x(t) = v_{xo} + at$$ $$x(t) = x_o + v_{xo}t + \frac{1}{2}at^2$$ where $$x_o = x(0)$$ is the position at $$t=0$$ and $$v_{xo} = v_x(0)$$ is the velocity at $$t=0$$.

### Derivation of the Constant Acceleration Equations

The following equations only apply for systems with a constant acceleration, $$a$$. We start by identifying two points in time we'll call $$t_i$$ and $$t_f$$, the initial time and the final time. From the position and velocity equations for constant acceleration, we can calculate the initial and final position and velocity: $$x_i = x(t_i)=x_o+v_{xo}t_i+\frac{1}{2}at_i^2$$ $$x_f = x(t_f)=x_o+v_{xo}t_f+\frac{1}{2}at_f^2$$ $$v_{xi} = v_x(t_i)=v_{xo}+at_i$$ $$v_{xf} = v_x(t_f)=v_{xo}+at_f$$ where $$x_o$$ and $$v_{xo}$$ are the position and velocity at $$t=0$$, which is not necessarily $$t_i$$. We also define: $$\Delta t = t_f-t_i$$, $$\Delta x = x_f-x_i$$, and $$\Delta v_x = v_{xf}-v_{xi}$$. The video shows how to derive the following relationships: $$v_{xf} = v_{xi} + a\Delta t$$ $$x_f = x_i + v\Delta t + \frac{1}{2}a(\Delta t)^2$$ $$v_{xf}^2 = v_{xi}^2 + 2a\Delta x$$ $$x_f = x_i + \frac{1}{2}(v_{xi}+v_{xf}) \Delta t$$

### Constant Acceleration: One Object, One Section

Here is a look at solving a constant acceleration starting with just one object and one section of constant acceleration. Time to start practicing and using your problem solving strategy: visualize, brainstorm, solve, and check. Review the module on this if necessary. As applied to constant acceleration problems, you might approach it like this:

1. Make sure you have large amounts of blank scratch paper.
2. Draw a picture.
3. Draw a schematic diagram with a coordinate system with a well-defined zero and positive direction
4. Clearly identify two points in time that you are interested in. Make sure the acceleration is constant between them.
5. Find and list what you know and what you need to know; including position, velocity, and acceleration for both points in time for all objects.
6. Use clear and unambiguous notation that creates meaning for you.
7. Determine what physics applies to the problem and find relationships between what you know and don't know.
8. If you have the same number of independent equations and unknowns you should be able to solve the problem.

### Constant Acceleration: One Object, Two Parts

What if the acceleration isn't constant? Well, I guess it shouldn't be in this module. But, wait! Sometimes you can encounter problems where the acceleration is not constant throughout the problem, but can be broken into sections where the acceleration is constant in each section. Special care needs to be taken because the number of parameters can start to become overwhelming. Make sure you spend proper time visualizing what is going on. Make sure you use notation and labels that make sense to you so you do not drown in symbols. One important point: for two sections, the final values from the earlier section become the initial values for the later section.

### Constant Acceleration: Two Objects

What happens if we add a second object? We must keep track of each object and apply the constant acceleration relationships to each object separately. Then, we use the context of the problem to find relationships between the objects. The physics isn't any different, but the notation can get busy! Go forward carefully, and use good pictures, schematics, and graphs.

### Using Graphical Methods

We talk about looking at problems using different representations. It helps you understand and visualize a problem. It helps you check that you got the right answer. But, I get it, to a lot of students it sounds like extra work. However, sometimes knowing how to use graphical methods can help you solve problems so much faster, you will want to apply them to every problem you encounter. I show you how in this video.

### Solution

1. You throw a ball straight up with an initial speed, v. What is its speed when it reaches half the maximum height it will attain?

### Solution

2. Looking out your window you see a ball pass right by your window going up. Your window is 1 m tall and the ball is visible for 1/4 second. How long will it be before the ball is visible again?

### Solution

3. When you are driving, if you had to stop quickly, you have a reaction time of 1/4 s. At 60 mph, you could stop in 50 m at max deceleration. How far would it take to stop going 75 mph?