Position, Velocity, and Acceleration

Position

Position x

Displacement

Initial Position xi

FinalPosition xf

Displacementx

Average Velocity

Starts here at a certain time

Stops here at a certain time

Average Velocity

Starts here at a certain time

Stops here at a certain time

More accurate the smaller the change is

Instantaneous Velocity

• Take the very small change in position over the very small change of time to be more accurate

• This will be equal to the slope of the position curve at a certain point

• Therefore is equal to the velocity at that point

dt

dx

Instantaneous Acceleration

• Same concept applies as velocity because acceleration is the change of velocity over time

• So the slope of the velocity equation give the acceleration at that point

• Therefore acceleration is equal to

t

v

dt

dv

Practice Problems

• The position of a particle is given by

• Since and

• And since and

342

9)(

2 xtS x)()(' tVtS 49)(' xtS

49)( xtV)()(' tAtV 9)(' tV

9)( tA

Going backwards

• Sometimes an acceleration or velocity equation will be given instead

• In that case, you will have to reverse differentiate, or integrate

• Solve for C each time you integrate before integrating again, with the given information.

Practice problem

Guidelines

• A(t) is the slope of the v(t) equation• Position is the integral of velocity, so

is equal to the displacement from the starting point at t=x

• Someone always turns around when the velocity graph goes from the 4th to 1st or 1st to 4th quadrant.

x

dttv0

|)(|