Motion in One Dimension

Displacement

A change in an object’s position: Δx = xf – xi

This is a vector quantity Units: m

How is displacement different from distance?

Average Velocity and Average Speed

€

v x =Δx

Δt

€

S =total d

Δt

Average Velocity from a Graph

t

x

A

B

x

t

€

vavg =Δx

Δt

Average Velocity from a Graph

t

xA

Bxt

€

vavg =Δx

Δt

Average Acceleration

€

ax =ΔvxΔt=vxf − vxit f − ti

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ax =Δ(Δx)

Δt

Average Acceleration from a Graph

t

v

A

B

x

t

€

aavg =Δv

Δt

t

x

Remember that the average velocity between the time at A and the time at B is the slope of the connecting line.

A

B

Instantaneous Velocity and Speed

Average Velocity from a graph

t

xA

B

What happens if A and B become closer to each other?

Instantaneous Velocity and Speed

Instantaneous Velocity and Speed

t

x

AB

What happens if A and B become closer to each other?

Instantaneous Velocity and Speed

t

x

AB

What happens if A and B become closer to each other?

Instantaneous Velocity and Speed

t

x

A

B

What happens if A and B become closer to each other?

Instantaneous Velocity and Speed

t

x

A

B

The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is represented by the slope of this tangent line.

A and B are effectively the same point. The time difference is effectively zero.

Remember that a limit is used to define a derivative by:

€

f '(x) = limΔx→0

f (x + Δx) − f (x)

Δx

Instantaneous Velocity and Speed

Instantaneous velocity is the limiting value of Δx/Δt as Δt approaches zero, or the derivative of x with respect to t:

The instantaneous speed of an object is the magnitude of its velocity €

vx = limΔt→0

Δx

Δt=dx

dt

Average and Instantaneous Acceleration

t

v

Average acceleration is represented by the slope of a line connecting two points on a v/t graph.

Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph.A

B

C

Instantaneous Acceleration

Instantaneous acceleration is the limiting value of Δv/Δt as Δt approaches zero, or the derivative of v with respect to t:

€

ax = limΔt→0

Δv

Δt=dv

dt

Acceleration can also be referred to as the second derivative of position with respect to time.

Just don’t let the new notation scare you; think of the d as a baby Δ, indicating a very tiny change!

€

ax =dv

dt=d2x

dt=Δ(Δx)

Δt

Now YOU try it!

Try these phun problems using the calculus

Derivatives: the graphical representation

Look closely at each of these graphs (all of which are for the same motion) x vs t, v vs t, and a vs t

Remember that derivatives have to do with slope (over an increasingly small interval)

Analyze each time interval and see if you can understand how these graphs are connected

Now YOU try it!

Using your knowledge of how position, velocity, and acceleration are connected, analyze and plot the following graphs

Draw representative graphs for a particle which is stationary.

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw representative graphs for a particle which has constant non-zero

velocity.

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw representative graphs for a particle which has constant non-zero

acceleration.

Given this graph of position vs time, plot the corresponding v vs t and a vs t graphs

Given this graph of velocity vs time, plot the corresponding x vs t and a vs t graphs