Momentum
The force and the time interval
Imagine a friend throwing you a ball. When you catch the ball,
the impact depends on both the mass of the ball and on its
velocity. We thus define the momentum of an object as
Would you rather change the momentum of your truck by
hitting a hay stack or running into a concrete wall ?
momentum = mass × velocity
p = mv
Δp = F
A bowling ball moving at a low speed and a golf ball moving at a
high speed can have the same momentum.
Δt
In the collision with the haystack, the time over which the
force acts is relatively long, so only a small force is needed to
change the momentum of the truck.
Changing the Momentum
Suppose the momentum of an object changes by an amount Δp.
Δp = Δ(mv) = m Δv
Δp =
If the velocity change Δv takes place over a time interval
Δt, then Δv = a Δt where a is the acceleration. Then if we
substitute this result into our equation for Δp, we have
Δp = m Δv = m (a Δt) = m a Δt
Now Newton tells us that the acceleration is due to a net
force F = ma. So putting this together with the equation for
Δp, we have
F
Δt
When the truck collides with the concrete wall, the collision
or interaction time is relatively short. Hence a large force
acts during this collision for the same momentum change.
List some other examples of the relationship between the
force used to change the momentum of an object and the time
interval over which the force acts.
Δp = m Δv = F Δt
The change in the momentum depends on both the force and
the time interval over which that force acts. The product
F Δt is called the impulse.
3.1
PHYS 1010Q
© DS Hamilton
A 60-kg passenger riding in a
car is involved in a 20 m/s (45 mi/h)
collision with a concrete barrier. If
the stopping time is 0.25 s, what is the
force exerted by the seat beat and shoulder strap on the
passenger?
3.2
PHYS 1010Q
© DS Hamilton
Energy
The principle of "Energy Conservation" says that the total
energy of an isolated system does not change. It can be
converted from one type of energy to another, but the total
amount of energy stays the same. It is one of the most
fundamental and far reaching concepts for how physics looks
at the universe around us.
Kinetic Energy
This is the energy associated with the motion of an object.
The kinetic energy is equal to one-half of the product of the
mass with the square of the speed,
KE = ½mv2
A tennis racket exerts a force F on a
50-g tennis ball to change its velocity
from 15 m/s north to 25 m/s south. The
contact time of the racket with the ball is
about 0.1 seconds.
The mks unit of energy is the Joule where 1J = 1kg·m2/s2.
What is the kinetic energy of an 80-kg sprinter running at a
speed of 10 m/s?
(a) What is the change in momentum of the tennis ball?
Notice that if the speed of an object is doubled, its kinetic
energy is quadrupled (22 =4).
(b) What is the force F on the ball?
What is the kinetic energy of the same sprinter running at
a speed of 20 m/s?
3.3
PHYS 1010Q
© DS Hamilton
3.4
PHYS 1010Q
© DS Hamilton
Work
The kinetic energy of an object can be changed when an
external force does work on that object. Work is defined as
the force in the direction of motion times the distance moved.
W =Fd
The prisoner may expend calories when he pushes
on the wall, but if the wall doesn't move, no work
is performed on the wall. In general, work can be
positive, negative, or zero.
Work and Kinetic Energy
The work done by a force can change the kinetic energy of an
object.
W = ΔKE
A 0.5-kg hockey-puck is initially at rest. What is its kinetic
energy after a net force of 0.4 N acts on it for a distance of
0.8 m?
Consider a girl who pushes a box with a 100 N
horizontal force that results in the box moving a
distance of 3 m along the floor. What is the
work done on the box by the girl?
A 80 N frictional force from the box sliding on the floor
opposes the motion. Find the work done on the box by this
friction force.
A toy car has an initial kinetic energy of 6 J. What is its
kinetic energy after a frictional force of 0.5 N has acted on
the car for a distance of 4m?
What is the work done on the box by the vertical
gravitational force as the box moves horizontally?
3.5
PHYS 1010Q
© DS Hamilton
3.6
PHYS 1010Q
Work and Potential Energy
What is the change in
PE of a 1-kg mass for
each case shown in the
figure?
We can also store energy when we do work.
Potential energy is "stored energy" and can
be used later to change an object's kinetic
energy. In this example the energy stored in
the bow is ΔPE = Fd, and is equal to the work
done by the archer.
When an object is lifted upward through a height
h, the gravitational potential energy increases by
mgh. (It decreases by mgh when the object is
lowered.) For this example, the work done by the
athlete on the barbell is stored as potential
energy, ie W = ΔPE.
How much work is needed to lift a bowling ball of weight
60 N and place it on a shelf 1.5 m above the floor?
© DS Hamilton
Work is done to lift the mass and increase
its gravitational potential energy. As the
mass falls, this potential energy is then
converted into the kinetic energy of
motion. The loss in potential energy just
equals the gain in kinetic energy.
Conservation of Energy
What is the change in the gravitational potential energy of
the bowling ball? What is the final potential energy of the
bowling ball when it on the shelf assuming than the potential
energy is chosen to be zero at the floor.
3.7
PHYS 1010Q
© DS Hamilton
Energy cannot be created or
destroyed; it may be transformed
from one form to another, but the
total amount of energy never changes.
Initially, the energy is all potential energy and is equal to mgh.
As the pendulum moves toward the low point, this potential
energy is converted to the kinetic energy of motion. On the
way back up, this kinetic energy is being converted back to
potential energy. The total energy stays constant as it the
pendulum swings back and forth.
3.8
PHYS 1010Q
© DS Hamilton
A circus diver has 10,000 J of potential
energy at the top of the pole. As he dives,
this potential energy is being converted into
kinetic energy. The total mechanical energy,
which is the sum of the kinetic and potential
energy, stays constant at each position along
the dive.
If the mass of the diver is 50 kg, what is
the height of the pole?
Potential energy can be stored in many different forms.
Gravitational PE
Elastic PE
Electrostatic PE
Chemical PE
Nuclear PE
bowling ball on high shelf
wound clock-spring
cloud in a thunderstorm
fire cracker, cheeseburger
uranium fission & hydrogen fusion
and then converted into the kinetic energy of motion.
Power
What speed did the diver have as he
entered the water bucket?
PHYS 1010Q
© DS Hamilton
Efficiency
The transformation of energy from one form to another never
happens with 100% efficiency. The efficiency can be defined
in terms of either energy or power conversion;
e=
useful energy output
useful power output
=
energy input
power input
A 60 W filament bulb and an 11 W compact fluorescent
lamp (CFL) both emit 9 W of visible light. Find the
efficiency of these two bulbs.
The “metabolic efficiency” of the human body is about 25%.
So for every Joule of food energy we consume, 0.75 J goes to
maintaining body temperature and the functioning of our
internal organs. The remaining 0.25 J is available for
mechanical work.
How high of a mountain can you climb using the
energy from a 200 Cal = 840 kJ candy bar? Assume
your mass is 80 kg.
3.11
PHYS 1010Q
ΔE
Δt
If you do one Joule of work every second, you are generating
power at a rate of one Watt.
P=
If a 60-kg sprinter can accelerate from a standing start to
a speed of 10 m/s in a time of 3 sec, what average power is
generated? (Hint: first find the change in the kinetic energy.)
A 2-kg object bounces down a set stairs
that are 3 m high. What is the kinetic
energy of the object when it reaches the
bottom?
3.9
Power is the rate that energy is transformed or transferred.
© DS Hamilton
3.10
PHYS 1010Q
© DS Hamilton