Kinematics

v=v_0 + at

Sometimes it is advantageous to write this with acceleration on one side and velocity on the other.

at=v-v_0

The next kinematic equation has expressions for position, velocity and acceleration.

x=x_{0} +v_{0}t + \dfrac{1}{2}at^2

Sometimes an author says distance instead of position. Some caution is needed. By itself, ‘x’ is not distance. We get a distance when we calculate the difference between two points. We are requiring x_0 when we say x is distance.

distance = x - x_0

There is a kinematic equation that doesn’t have the variable time in it. We can derive this using the two kinematic equations listed above. To make the math simpler, we will assume that the position at time zero, is zero, and the velocity at time zero is zero.

v = v_0 + at

v = at

x = x_0 + v_0 t + \dfrac{1}{2}at^2

x = \dfrac{1}{2}at^2

We can replace time with velocity divided by acceleration:

t = \dfrac {v} {a}

x = \dfrac{1}{2}a \dfrac {v^2} {a^2}

x = \dfrac {v^2} {2a}

v = \sqrt {2ax}

Don’t panic if you see ‘d’ instead of ‘x’. Some equations prefer ‘d’ since ‘x’ is measuring the distance traveled under the constant acceleration.

If you are feeling feisty, go back and do the math allowing for x_0 to be nonzero and then go back and also allow t_0 to be nonzero.