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Velocity and acceleration
Speed is a scalar
elapsed time
traveleddistance speed avg
Note: Speed is always positive.
avg is short for average 平均
1400 km
What is the distance traveled in meters?
The flight took 2 hours and 20 minutes. What is this in seconds?
Velocity is a vector
elapsed time
nt vectordisplaceme velocity avg
Units: m/s
速度
Velocity is a vector
t
rv
avg
Remember: Δ is the Greek letter Delta, and means ‘change in something’.
Dimensions on both sides must match
tr
v
avg
1T L 1T LT
L
1400 km
x
y
x
50x
y40y
Calculating a unit vector from angles
zyxr cos ,cos ,cosˆ
where θ is the angle between the vector and the relevant axis.
Instantaneous velocity
t
r
t
rv
t d
dlim
0inst
Our first use of calculus.
微积分
瞬时
Instantaneous velocity
t
r
t
rv
t d
dlim
0inst
“the derivative of r with respect to t”
导数
Instantaneous velocity
t
r
t
rv
t d
dlim
0inst
“the rate of change of r”
The derivative of a vector is actually three derivatives
dt
d,
dt
d,
dt
d,,
dt
d
d
d zyxzyx
t
r
A reminder…
1
23
2
d
d
3d
d
2d
d
1d
d
nn
nxx
x
xx
x
xx
x
x
x
What is the solution to
?
(1) (2) (3) (4) (5)
135.02dt
d 2 tt
13
5.04
1325.076.0
265.04
5.02
23
2
t
ttt
t
t
Acceleration: rate of change of velocity
加速度
elapsed time
yin velocit changeon accelerati avg
Units: m/s2
Acceleration: rate of change of velocity
t
va
avg
t
v
t
va
t d
dlim
0inst
Average acceleration:
Instantaneous acceleration:
Bumper car ride
Bumper car ride
Example: The position of a bumper car is described as a function of time by:
m 0 ,210 ,5.052.0 22 ttttr
Find:
(a) The displacement and average velocity between t = 1.0 s and t = 3.0 s.
Bumper car ride
Example: The position of a bumper car is described as a function of time by:
m 0 ,210 ,5.052.0 22 ttttr
Find:
(b) The instantaneous velocity of the car at t = 3.0 s.
Bumper car ride
Example: The position of a bumper car is described as a function of time by:
m 0 ,210 ,5.052.0 22 ttttr
Find:
(c) The instantaneous acceleration of the car at t = 3.0 s.
