L-4 Free fall

1
L-4 Free fall constant acceleration

• Galileo In the absence of air resistance, all
  objects, regardless of their mass, fall to earth
  with the same acceleration (change in velocity) g
• g ? 10 m/s2 (10m/s)/s ? the speed of a falling
  object increases by 10 m/s every second.
• This means that if 2 objects start at the same
  height, they will hit the ground at the same
  time.
• This is easier to see on an
• inclined plane, where the
• effect of gravity issmaller down the plane.

2
Free fall velocity and distance
time(s) speed(m/s) distance(m) d(m)
0 0 0
0.45 4.5 1
1 10 5 ½10(1)2
2 20 20 ½10(2)2
3 30 45 ½10(3)2
4 40 80 ½10(4)2
5 50 125 ½10(5)2

• If you drop a ball from the top of a building it
  gains speed as it falls.
• Every second, its speed increases by 10 m/s.
• Also it does not fall equal distances in equal
  time intervals

Galileo recognized this pattern
3
effect of air resistance terminal velocity
? air resistance increases with speed ?
m 100 kg, Fgrav w mg 100 kg ?10 m/s2
1000 N
A person who has their hands and legs
outstretched attains a terminal velocity of about
125 mph.
4
Motion with constant acceleration

• A ball falling under the influence of gravity is
  an example of motion with constant acceleration.
• acceleration is the rate at which the velocity
  changes with time (increases or decreases)
• acceleration is the change in meters per sec per
  second, so its measured in m/s2 or ft/s2 or
  mph/s
• if we know where the ball starts and how fast it
  is moving at the beginning we can figure out
  where the ball will be and how fast it is going
  at any later time!

5
Simplest case constant velocity ? acceleration
0

• If the acceleration a 0, then the velocity is
  constant.
• In this case the distance an object will travel
  in a certain amount of time is given by
  distance velocity x time (a0)
• d v ? t (for a 0 only)
• For example, if you drive at 60 mph for one hour
  you go 60 mph x 1 hr 60 mi.

6
Example running the 100 m dash

• Usain Bolt set a new world record in the 100 m
  dash at9.58 s! Did he run withconstant
  velocity, or washis motion accelerated?
• He was not moving in theblocks (at rest), then
  he began moving when the gun went off, so his
  motion was clearly accelerated
• Although his average speedwas about 100 m/10 s
  10 m/s, he probably did not maintain this speed
  all through the race.

7
running the 100 m dash
100 m
speed
distance
start
Finish line
the winner has the highest average speed 100 m
/ time
8
100 m dash (Seoul 1988)
9
constant acceleration

• Example Starting from rest, a car accelerates up
  to 50 m/s (112 mph) in 5 sec. Assuming that the
  acceleration was constant, compute the
  acceleration.
• Solution acceleration (a) rate of change of
  velocity with time

10
The velocity of a falling ball

• Suppose that at the moment you start watching the
  ball it has an initial velocity equal to v0
• Then its present velocity (v) is related to the
  initial velocity and acceleration (a) by
• present velocity
• initial velocity acceleration ? time
• Or in symbols v v0 a ? t (for a
  constant)
• v0 is the velocity when the clock starts (t0)
  and v is the velocity at time t later

11
Ball dropped from rest

• If the ball is dropped from rest, that means that
  its initial velocity is zero, v0 0
• Then its present velocity a ? t, where a is the
  acceleration of gravity, which we call g ? 10
  m/s2 or 32 ft/s2, for example
• What is the velocity of a ball 5 seconds after it
  is dropped from rest from the top of the Sears
  Tower (now the Willis Tower)?
• ? v 32 ft/s2 ? 5 s 160 ft/s (109
  mph)

12
The position of a falling ball

• Suppose we would like to know where a ball would
  be at a certain time after it was dropped
• Or, for example, how long would it take a ball to
  fall to the ground from the top of the Sears
  Tower (1450 ft).
• Since the acceleration is constant (g) we can
  figure this out!

13
Falling distance

• Suppose the ball falls from rest so its initial
  velocity is zero
• After a time t the ball will have fallen a
  distance distance ½ ? acceleration ?
  time2
• or d ½ ? g ? t2 (g 10 m/s2)

14
Falling from the Sears Tower

• After 5 seconds, the ball falling from the Sears
  Tower will have fallen distance ½ ? 32 ft/s2
  ? (5 s)2 16 ? 25 400 feet.
• We can turn the formula around to figure out how
  long it would take the ball to fall all the way
  to the ground (1450 ft)? time square root of
  (2 x distance/g)

15
Look at below!

• or
• when it hit the ground it would be moving at v
  g ? t 32 ft/s2 ? 9.5 sec 305 ft/s
• or about 208 mph (watch out!)

16
How high will it go?
v 0 for an instant

• Lets consider the problem of throwing a ball
  straight up with a speed v. How high will it go?
• As it goes up, it slows down because gravity is
  pulling on it.
• At the very top its speed is zero.
• It takes the same amount of time to come down as
  it did to go up.
• going down v vo gt, where, vo 0, so v gt
  ? t v/g
• going up tup vo/g

vo
17
Example

• A volleyball player can leapup at 5 m/s. How
  long is shein the air?
• SOLUTION? total time ttotal tup tdown
• time to get to top tup vo / gwhere vo is
  the initial upwardvelocity
• tup 5 m/s / 10 m/s2 ½ sec
• ttotal ½ s ½ s 1 s

18
An amazing thing!

• When the ball comes back down to ground level it
  has exactly the same speed as when it was thrown
  up, but its velocity is reversed.
• This is an example of the law of conservation of
  energy.
• We give the ball some kinetic energy when we toss
  it up, but it gets it all back on the way down.

19
So how high will it go?

• If the ball is tossed up with a speed v, it will
  reach a maximum height h given by
• Notice that if h 1m,
• this is the same velocity that a ball will have
  after falling 1 meter.

20
Escape from planet earth(Not everything that
goes up must come down!)

• To escape from the gravitational pull of the
  earth an object must be given a velocity at least
  as great as the so called escape velocity
• For earth the escape velocity is 7 mi/sec or
  11,000 m/s, 11 kilometers/sec or about 25,000
  mph.
• An object given this velocity (or greater) on the
  earths surface can escape from earth!

21
Formulas apply whenever there is constant
acceleration - example

• a car moving at v0 3 m/s begins accelerating at
  a 2 m/s2. When will its velocity increase to 13
  m/s?
• SOLUTION
• v v0 a ? t
• 13 m/s 3 m/s 2 m/s2 ? t
• 13 m/s 3 m/s 10 m/s
• ? t 5 seconds

22
Example deceleration slowing down

• deceleration means that the acceleration is
  opposite in direction to the velocity
• Suppose you are moving at 15 m/s and apply the
  brakes. The brakes provide a constant
  deceleration of 5 m/s2. How long will it take
  the car to stop?
• v v0 a t
• 0 15 m/s (5m/s2) t
• ? t 3 s

23
Another example

• To spike the ball, a volleyball player leaps 125
  cm straight up.
• What was her speed when she left the court?
• formula ?
• 125 cm 1.25 m