LINER MOMENTUM
Impulse and momentum
The momentum of a body is defined as the product of its mass m and its velocity. Unit kgm/s
𝑃 = 𝑚𝑣
Momentum is a vector quantity, it is called linear momentum to distinguish it from angular momentum
The concept of impulse is actually associated with collision, when a body collide with another, it
received a blow of impulse. The impulse consist of a large forces acting for a short time.
Impulse can be defined as the product of the average force acting on a particle and the time during
which it acts.
𝐼 =𝐹×𝑡
Example: a stationary ball is hit by an average force 50N for a time 0.03sec. what is the impulse
experience by the body.
Example: a body of mass 3.0kg moves with a velocity 10m/s calculate the momentum of the body
NEWTON’S LAW OF MOTION
Motion is caused by unbalanced force, how force are related to motion was first discovered by Isaac
Newton’s, who also stated three laws of motion, known as Newton’s law of motion.
The first law states that every object continues in its state of rest or of uniform motion in a straight
line unless acted upon by an external force.
Inertia is the tendency of a body to remain at rest or uniform linear motion in the absence of applied
force. Inertia is a property of matter.
Newton’s first law explain that
1. What Force does, but does not suggest how force should be measure
2. It also explain why a passenger in a fast moving car tends to move forward when the cars
suddenly stops
Newton’s second law state that the rate of change of momentum is proportional to the impressed
force and take place in the direction of the force.
It enables us to define an absolute unit of force which remains constant under all conditions
𝐹𝛼 =
𝑐ℎ𝑛𝑎𝑔𝑒 𝑖𝑛 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟 𝑡ℎ𝑎𝑡 𝑐ℎ𝑎𝑛𝑔𝑒
𝑚𝑣 − 𝑚𝑢
𝑡
𝑣−𝑢
𝐹𝛼 = 𝑚 (
)
𝑡
𝐹𝛼 =
𝐹𝛼 = 𝑚𝑎
𝑣−𝑢
(
)=𝑎
𝑡
Thus
𝑓 = 𝑘𝑚𝑎
K=constant
Unit of force is chosen so that k=1 hence
𝑓 = 𝑚𝑎
And
𝑣−𝑢
𝐹 = 𝑚(
)
𝑡
𝑓. 𝑡 = 𝑚𝑣 − 𝑚𝑢
Example: an unbalanced force of 20N acts on a 4.0kg mass. What acceleration does it give
Example: a 900g stone is pushed along a tarmac by a horizontal force of 2N. A frictional force of 8N
opposed the motion. What is the acceleration given to the stone
Example: A ball of mass 0.3kg moving at a velocity of 20m/s is suddenly hit by a force of 5N for 0.03sec.
Find the velocity of the motion
Example: a body of mass 0.1kg dropped from a height of 8m onto a hard floor bounce back to a height
of 2m. Calculate the change in momentum if the body is in contact with the floor for 0.1sec. What is the
force exerted on the body
Newton’s third law: of motion state that action and reaction are equal and opposite or to every action
there is an equal and opposite reaction.
The law implies that when a body A exert a force Fa on a body B, the body B always exert a force Fb on
the body A.
Both force are equal in magnitude but opposite in direction
Hence FA=-FB
CONSERVATION OF LINEAR MOMENTUM
The principle of linear momentum states that
1. In any system of colliding objects the total momentum is always conserved provided that there
is no net external force acting on the system
2. The total momentum of an isolated or closed system of colliding bodies remain constant
3. If two bodies collide in a closed system, the total momentum after the collision is equal to the
total momentum before collision.
By a closed or isolated system we mean that system on which no external force act
If
FA=-FB
mAaA=- mBaB
but
𝑣−𝑢
𝑎=(
)
𝑡
Hence
𝑣−𝑢
)=𝑡
mA(
𝑣−𝑢
)
𝑡
mB(
eliminating t
(mAVA-mAuA)=-(mBvB-mBuB)
(mAuA-mAuA)=-(mBvB-mBvB)
Momentum = mass X velocity
It means that, the momentum before collision equal momentum after collision
Example: A bullet of mass 0.05kg is fired horizontally into 10kg block which is free to move. If both
bullet and block move with velocity 0.5m/s after impact, find the velocity with which the bullet hit the
body.
Example: A body(p) of mass 5kg moving with velocity of 30m/s collides with another body Q, moving in
opposite direction with a velocity of 20m/s. if both bodies now move in the direction of p at velocity of
10m/s. calculate the mass of Q.
COLLISION
Collision is a short duration interaction between two bodies or more than two bodies simultaneously
causing a change in motion of the bodies, or any event in which two or more bodies exert forces on each
other in a relatively short time.
There are two types of collision
Elastic collision
Inelastic collision
Elastic collision: the collision between two bodies, the k.e is the same before and after the collision
We can write the equation between the law of conservation of momentum and the law of conservation
of k.e
Example of elastic collision
1. A moving cue ball hit a resting ball
2. Collision of billard balls
3. Collision of molecules of atoms
In elastic collision, the relative velocity of the two bodies is uncharged in magnitude but reversed in
direction.
Inelastic collision: is a collision which k.e is not conserved due to the action of internal friction there is
loss of k.e while momentum is conserved k.e is not conserved
Example
1. Collision of a high speed car with an object
2. Dropping of ball of clay
For a completely in elastic collisions, the k.e before collision is greater than the k.e after collision
Example: two bodies A and B of masses 4kg and 2kg move towards each other with velocity 3m/s
and 2m/s and collide. If the collision is perfectly elastic, find the velocity of the two bodies after
collisions. Find the total kinetic energy of the system before and after collision, hence calculate loss
in kinetic energy
Applications of newton’s and conservation of momentum laws
1. Recoil of a gun
2. Jet and rocket propulsion
3. Why walking is possible
Example: a rifle of mass 15kg fires a bullet of mass 60g with a velocity of 200m/s. calculate the velocity
of the riffle.
Example: a rocket is burning fuel at the rate of 200g/s and ejecting all the gas in one direction at the rate
of 400m/s. what is the maximum Weight the rocket can have if it is going to move vertically upwards.
MCQ
1. A ball of mass 0.5kg moving at 10m/s collides with another ball of equal mass at rest. If the two
balls move off together after the impact, calculate their common velocity.
(A). 0.2m/s
(B). 0.5m/s
(C). 5.0m/s
(D). 10m/s
2. A ball of mass 6.00kg moving with a velocity of 10.0m/s collide with a 2.0kg ball moving in the
opposite direction with a velocity of 5.0m/s. after collision the two ball coalesce and move in the
same direction. Calculate the velocity of the composite body.
(A). 5.0m/s
(B). 6.25m/s
(C). 8.75m/s
(D). 12.0m/s
3. A constant force of 5N acts for 5 seconds on a mass of 5kg initially at rest. Calculate the final
momentum
(A). 124kgm/s
(B). 25kgm/s
(C). 15kgm/s
(D). 5kgm/s
4. When taking a penalty kick, a footballer applies a force of 30.N for a period of 0.005s. if the mass
of the ball is 0.0075kg, calculate the speed with which the ball fall off
(A). 4.0m/s
(B). 11.25m/s
(C). 20.0m/s
(D). 45.0m/s
5. A body of mass 100g moving with a velocity of 10.0m/s collides with a wall. If after the collision,
it moves with a velocity of 2.0m/s in the opposite direction, calculate the change in momentum
(A). 0.8Ns
(B). 1.2Ns
(C). 12.0Ns
(D). 80.0Ns
6. For elastic collision
(A). Energy is doubled and momentum id halved
(B). Energy is conserved
(C). Momentum is conserved
(D). Kinetic energy and momentum are conserved
7. The property of a body to reamin at rest or to continue to move in a straight line, is known as
(A). Force
(B). Impulse
(C). Momentum
(D). Inertia
8. In an elastic collision,
I.
Energy is conserved
II.
Energy is decreased
III.
Energy is increased
IV.
Linear momentum is conserved
(A). I only
(B). II only
(C). III only
(D). I and IV only
9. What should you do to reduce the amount of effort needed to lift something using a first class
lever?
A. move the fulcrum to the middle of the lever
B. move the fulcrum closer to the load
C. move the fulcrum closer to the effort
D.reduce the size of the fulcrum
10. The efficiency of a simple machine is ____________________________.
A. is always less than 100%
B. is equal to 100%
C. is always 50%
D. is always more than 100%.
11. If the mechanical advantage of a simple machine is 4, then the
A. output force is 4 times the effort
B. effort is 4 times the output force
C. efficiency is 4%
D. the work output is 4 times the input
Use Figure 1 to answer questions 12-13.
12. Which of the following statements is true for Figure 1.
A. B – is the fulcrum, C- is the resistance, A – is the effort
B. B – is the resistance, C – is the fulcrum, A – is the effort
C. B – is the fulcrum, A – is the effort, C- is the resistance
D. B – is the resistance, A – is the fulcrum, C – is the effort
13. In Figure 1, if the distance from a to b is 20 cm, and the distance from a to c is 80 cm, then the
mechanical advantage of the system is
A. 20
B. 80
C. 4
D. ¼
14. If you are using a screwdriver to pry open a paint can, you are using it as what type of simple
machine?
(A). Lever
(B). Pully
(C). Screw jack
(D). Wheel and axle
Theory
1. States Newson’s law of motion. Derive from one of the laws show the relationship between
the momentum produced in a body and the force applied on the body
2. State the law of conservation of linear momentum. A 3.00kg riffle lays on a smooth table
when it suddenly discharges, firing a bullet of 0.02kg with a speed of 500m/s. calculate the
recoil speed of the gun.
3. Distinguished between impulse and momentum and according to newton’s law of motion
what is the relationship between impulse and momentum
4. Calculate the momentum of
a. The moon of mass 7x1022kg travelling with a velocity of 1.05km/s
b. A tanker of mass 7x107kg travelling at 4m/s
c. An aircraft of mass 2.5x105kg travelling at 500m/s
5. Distinguished between
a. Elastic and inelastic collision
b. Inertia mass and weight
6. Derive from newton’s law the relationship between mass and acceleration, A 15.0kg
monkey hang from a cord suspended from the ceiling of an elevator. The cord can withstand
a tension of 200N and breaks as the elevator accelerates. What was the elevator minimum
acceleration in magnitude and direction?