PhysicsPhysicsChapter 5Chapter 5
Position-Time GraphPosition-Time Graph Time is always on the x axisTime is always on the x axis The slope is speed or velocityThe slope is speed or velocity
Time (s)
Pos
ition
(m
)Slope = Δ y
Δ x
Velocity-Time GraphVelocity-Time Graph Time is always on the x axisTime is always on the x axis The slope is accelerationThe slope is acceleration Area under the curve is positionArea under the curve is position
Slope = Δ y Δ x
time (s)
Vel
ocity
(m
/s)
Area under velocity time graph is positionArea under velocity time graph is position
Time (s)
Vel
ocity
(m
/s)
Area = ½ b * hArea = ½ b * h
For this triangleFor this triangle
A = ½ (velocity) (time)A = ½ (velocity) (time)
Acceleration -Time GraphAcceleration -Time Graph Time is always on the x axisTime is always on the x axis Area under the curve is velocityArea under the curve is velocity
time (s)
Acc
eler
atio
n (m
/s/s
)Slope = Δ y
Δ x
Area under Area under acceleration time graph is velocityacceleration time graph is velocity
Time (s)
Acc
eler
atio
n (m
/s/s
)
Area = ½ b * hArea = ½ b * h
For this triangleFor this triangle
A = ½ (acceleration) (time)A = ½ (acceleration) (time)
Acceleration is often Acceleration is often graphed like thisgraphed like this
time (s)
Acc
eler
atio
n (m
/s/s
)
l+
Which makes area Which makes area under the curve …under the curve …
time (s)
Acc
eler
atio
n (m
/s/s
)
l+
Area = b * hArea = b * h
For this For this
A = (acc) (time)A = (acc) (time)
Looking at graphsLooking at graphs Average uses slope of the chordAverage uses slope of the chord Instantaneous uses slope of the Instantaneous uses slope of the
tangenttangent If slope of the chord = slope of If slope of the chord = slope of
the tangent line then average = the tangent line then average = instantaneousinstantaneous
AverageAverageVelocityVelocity
01
01
tt
dd
t
dv
101 tvdd
Which leads to a Kinematic Equation
01
01
tt
ddv
101 tvdd
0101 ddttv
011 ddtv Let time at 0 be 0Let time at 0 be 0
oror
vtdd 0
Position with Position with Constant VelocityConstant Velocity
AverageAverageaccelerationacceleration
01
01
tt
vv
t
va
atvv 0
Which leads to another Kinematic Equation
01
01
tt
vva
atvv 0
0101 vvtta
011 vvta oror
Let time at 0 be 0Let time at 0 be 0
Final position withFinal position withConstant accelerationConstant acceleration
tvvdd 00 21
tvvd 021
Time (s)
Vel
ocity
(m
/s)
d = vd = v00 * t * t
vv00
vv
tt
d = ½ (v – vd = ½ (v – v00) * t) * t
oror
d = ½ vt – ½vd = ½ vt – ½v00tt
Add them together Add them together & you get& you get
If the initial position is not If the initial position is not zero, then add dzero, then add d00 to the to the
total distancetotal distance
tvvdd 00 21
Final position withFinal position withConstant accelerationConstant acceleration
200 2
1 attvdd
200 2
1 attvdd
If v is not known, If v is not known, substitute the substitute the
following equation following equation in for vin for v
tvvdd 00 21
atvv 0
tatvvdd 000 21 This leads This leads
to…to…
Final velocity withFinal velocity withConstant accelerationConstant acceleration
020
2 2 ddavv
advv 220
2 Or simplyOr simply
Final velocity withFinal velocity withConstant accelerationConstant acceleration
020
2 2 ddavv
advv 220
2 Or simplyOr simply
To solve this To solve this equation, note equation, note that it does not that it does not include time.include time. tvvdd 00 2
1
020
2 2 ddavv
atvv 0
a
vvvvdd 0
00 21
a
vvt 0
Solve for tSolve for t
Sub t into:Sub t into:
a
vvvvdd
200
0
20
202 vvdda
Kinematic EquationsKinematic Equations
vtdd 0 atvv 0
tvvdd 00 21
200 2
1 attvdd
020
2 2 ddavv
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