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Vertical Motion
Circular Motion
Vector’s Components Velocity has both the magnitude and direction (is a vector).
For motion we need to know: distance, time, speed, velocity, acceleration.
2
The average speed is the total distance divided by the total time or change in
distance divided by change in time:
Instantaneous speed is change in distance divided by very short time
(t ! zero):
Both average and instantaneous speed have only a magnitude (is a scalar).
totaltotalavtdv /=
0/ !"""= ttdvinst
tdvav
!!= /
The acceleration shows how fast the object changes its velocity;
the change in velocity divided by the change in time: tva !!= /
The distance traveled by an object at constant acceleration: d = v0t +
at2
2
3
Types of Motion
1. Horizontal motion
2. Vertical motion
3. Circular motion
(a) Motion at constant speed
(b) Motion at constant acceleration
constv =
consta =
4. Projectile motion
v!
v!
v!
v!
v!
v!
A car traveled 40 m for 4 seconds at constant acceleration from the rest.
Find the acceleration of the car.
Distance and Acceleration
1) 2 m/s2
2) 4 m/s2
3) 5 m/s2
4) 10 m/s2
4
2
22/5
16
80
)4(
)40)(2(sm
s
m
s
m===
2
2at
d =
2
2
t
da =
" An extremely important case of constant acceleration motion.
Vertical Motion, Free Fall
d =gt
2
2
" This means that any two objects, regardless of the mass or material
of the objects, released from a given height will take the same timeto reach the floor.
" When an object is released in free fall, it falls down with a constant
acceleration a = g = 9.8 m/s2 which is always directed down! (NB weare assuming here that air resistance is negligible)
Distance for free fall:
v = gtVelocity for free fall:
5
Falling Object
If you dropdrop a ball, it will have:
– zero initial velocity;
– the acceleration of 9.8 m/s2
due to gravity.
+
289 m/s.ga ==
6
The velocity and position can be
written as:
2
2gtd =gtv =
The velocity and acceleration have
the same direction (downward).
Throwing Object Upward
If you throwthrow a ball upward, it will have:
– a non-zero initial velocity;
– the acceleration of 9.8 m/s2 due to
gravity.
7
+
gtvv !=0
2
2
0
gttvd !=
At the highest point of elevation, the speed
is zero while the acceleration is 9.8 m/s2 !
The velocity and acceleration have the opposite
direction, upward and downward, respectively.
The ball is dropped from the 40 m high building.
Falling Ball
1. Find its speed after 2 seconds:
2. How long it takes time to reach the ground?
3. Find its speed near the ground?
8
smssmgtv /6.19)2)(/8.9( 2===
2
2gtd =
22 gtd = s
sm
m
g
dt 86.2
/8.9
)40)(2(22===
smssmgtv /28)86.2)(/8.9( 2===
A ball is thrown upward with the initial speed of 25 m/s.
Ball Throwing Upward
2. Find speed of the ball after 2 seconds:
1. How long it takes time to reach the highest point?
3. Find the distance from the ground to the highest point:
9
gtvv !=0
mssm
ssm 89.312
)55.2)(/8.9()55.2)(/25(
22
=!=
0== hpvv ssm
sm
g
vt 55.2
/8.9
/25
2
0===
gtvv !=0
2
2gttvd o !=
smssmsm /4.5)2)(/8.9(/25 2=!=
gtv =0
Circular Motion: Centripetal (Radial) Acceleration
This type of acceleration occurs when an
object moving on a circular path.v
v
v
v
rada
rada
rada
10
rada
rada
rada
rada
v
v
v
v
Acceleration is directed to the center, so-
called “centripetal acceleration” (the word
centripetal is derived from two Greek words
“seeking the center”) or “radial acceleration”.
The magnitude of the centripetal (radial)
acceleration: r
vaa radlcentripita
2
==
courve theof radius speed, == rv
r
r
v
rada
Circular Motion: Period and Frequency
fTor
Tf
11==
11
In a circular motion the time required for one rotation is the period, T
The unit of period: second (s)
For a circular motion or other periodic process we also can use the frequency, f
A wheel makes one revolution during 0.02 s. Find the frequency of its
rotation.
HzssT
f 501
5002.0
11====
The unit of frequency: 1/second = hertz ( Hz)
Centripetal (Radial) Acceleration
A ball rotates on a circle with a radius of 2 m at a speed of 4 m/s. Its
centripetal (radial) acceleration is:
1) 2 m/s
2) 2 m/s2
3) 8 m/s
4) 8 m/s2
22
/82
)/4(sm
m
sm==
r
vacentr
2
=
)radius(r
)velocity(v!
)onaccelerati
radialor lcentripeta(a!