# Kinematic Equations of Motion

1. Mar 18, 2009

### randeezy

Can anyone help me familiarize myself with the Kinematic Equations of Motion?

I need to know them for this summer institute.

Thanks,

Randeezy

2. Mar 18, 2009

### Staff: Mentor

Last edited by a moderator: May 4, 2017
3. Mar 18, 2009

### randeezy

Thank you, but is it possible that you could explain to me what each is specifically used for, and how to use them?

4. Mar 18, 2009

### Feldoh

I'll take a shot...

The branch of kinematics deals with motion without focusing on the causes of the motion. Generally when we talk about motion we talk about a few different quantities.

When we talk about motion we are talking about quantities that have a direction and a magnitude associated with them. These quantities are called vectors. Quantities that just have magnitude are called scalars.

In kinematics we are primarily concerned with the vectors, position, velocity, and acceleration.

Position deals with the location of something in space. If we take cartesian coordinates and say you are located at (3,4). Well you've just defined a direction (3 units on the x axis, and 4 units on the y axis) and a magnitude (by the Pythagorean theorem the magnitude is 5). Usually position is denoted by x.

Velocity deals with a change in position. The scalar version of velocity is usually called speed. So with velocity we're concerned with a change in distance per unit time and the direction(s) in which this change occurs. Usually we denote velocity with v. The relationship between velocity and position can be seen by:

$$v_{ave} = \frac{\Delta x}{\Delta t}$$ where ave just denotes the average.

Acceleration is a change in velocity of a change of a change in position, if you'd like to think of it in that way. We denote acceleration usually as a.

$$a_{ave} = \frac{\Delta v}{\Delta t}$$.

In basic kinematics we usually only focus on problems involving constant acceleration, otherwise we have to discuss this with calculus. In fact all the equations I'm about to list are easily derived using differential and integral calculus but I'll try to explain them without all that.

Picture this example: You are walking in the positive x direction (still in Cartesian coordinates) with a constant velocity. Notice it's stated as constant velocity. It's constant in the sense that the velocity is not changing which means that there is no acceleration in this example. At some instant in time how do know where you are?

The relation $$\Delta x = vt$$, where $$\Delta x = x_{final}-x_{initial}$$ tells us where you are.

If you start walking at some x_i and want to know where you ended up we can rearrange this equation to give:

$$x_{final} = x_{initial} + v_{ave}t$$ (1)

By a similar argument it can be shown that if we have constant acceleration that the velocity is given by:

$$v_{final} = v_{initial} + at$$ (2)

Now if we have constant acceleration over some time interval then the velocity is changing so we can consider an average velocity over that time interval. This is given by:

$$v_{ave} = \frac{v_{final} + v_{initial}}{2}$$ (3)

Now if we sub equation (3) into equation (1) we get:

$$x_{final} = x_{initial} + v_{ave} t = x_{initial} + \frac{v_{final} + v_{initial}}{2}$$ (4)

Now if we sub equation (2) into equation (4) and some rearranging we get:

$$x_{final} = x_{initial} + v_{initial} t + (1/2) a_{ave} t^2$$ (5)

and

$$v_{final}^2 = v_{initial}^2 + 2 a \Delta x$$ (6)

This equations make up the basics of constant acceleration kinematics...

Equations 1-6 can be used to solve any problem that can be given to you. The reason for so many equations is because each equation is limited to certain quantities. For example it you wanted to find the final velocity knowing the initial velocity, acceleration, and the displacement you could use equation 6 to easily calculate it. Notice equation 6 is time independent. The other equations are listed because they are also independent of one of the quantities typically used.

5. Mar 18, 2009

### Staff: Mentor

6. Mar 18, 2009

### randeezy

Thanks. I'll see if I can wrap my head around it....

Maybe if I slept with it under my pillow.... lol