Position, Velocity, and Acceleration

Allow me to say the same thing about position and velocity in 5 different ways.

1. Velocity is the derivative of position .

2. Velocity is the instantaneous rate of change of position with respect to time .

3. An object's velocity tells us how much it's position will change when time changes.

4. Velocity at some time is just the slope of a line tangent to a graph of position vs. time

5. v(t) = \frac{dx}{dt} = \dot{x}(t)

It turns out you can say the same 5 things about velocity and acceleration .

1. Acceleration is the derivative of velocity .

2. Acceleration is the instantaneous rate of change of velocity with respect to time .

3. An object's acceleration tells us how much it's velocity will change when time changes.

4. Acceleration at some time is just the slope of a line tangent to a graph of velocity vs. time

5. a(t) = \frac{dv}{dt} = \dot{v}(t)

We can also make a couple interesting statements about the relationship between position and acceleration :

1. Acceleration is the second derivative of position .
2. a(t) = \frac{d^2}{dt^2}x(t) = \frac{d^2x}{dt^2} = \ddot{x}(t)

We'll explore this more in the next lesson. For now, just know that differentiating position twice gives acceleration!