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
€
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
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