Impulse and Momentum Theorem

The Meaning of linear momentum and Impulse. Rocket launching is a spectacular example of the application of the law of momentum conservation. The rocket gains propulsion by burning and ejecting its fuel's mass. At a certain hight, the rocket also gains velocity increase by releasing its fuel tanks. What is momentum? what is its relationship with mass and velocity

When a billiards game is started, the white ball is shot hard so that is moves and collides with the other set of balls. The other balls the move to several directions with different velocities. Some are moving to the right, left, slanting to the right, to the left, etc. some are moving fast, some are moving slowly, some even stay at rest. Why?

What are the factors made the balls to move in different directions and at different speeds? To answer such question, we should learn about linear momentum, Impulse, and the law of momentum conservation.

Impulse and Momentum Theorem

Linear momentum (often simply called momentum) of a body is defined as the product of the body's mass and its velocity. momentum, symbolized by p, is a vector quantity. Mathematically, it is written

P = mv,

where m is the mass of the body and v is the velocity.

The SI unit of momentum is kg.m/s. the direction of momentum is the same as the direction of its velocity.

In daily lives, according to equation (4.1), a body with a certain mass, which is moving at high velocity, will have greater momentum than another body with the same mass but is moving at a lower speed. On the other hand, two bodies that are moving with the same velocity but having different masses will also have different momentums. For example, a container truck will have a greater momentum than a sedan, even though they both are moving with the same velocity.

If force F is applied to a body with mass m, then Newton's second law holds F = ma. However,
a = Δv/Δt  so that we get;
Where Δp represents a change in momentum.
Equation (4.2) is another statement of the Newton's second law are have discussed before. It states that.

The force working on a body equal to the change of momentum per unit interval of time.

Equation [4.2] can be applied in the case where the mass of the body is changing toward the time, for example; in a motion of a rocket, which loses it mass when the fuel is burning out.
We could apply the concept of momentum change to a body where the applied force is not constant. For example is momentum in a ball, which is rolling at a certain velocity, and then kicked off in such a way that its velocity changing from to. In this situation, Equation (4.2) could be used to determine the average force working on the ball, according to the formula:

The equation above can be expressed as

P Δt = ΔP = P_t - P₀ = mv_t - mv

The product between F  and Δt is called impulse, denoted by I. Thus,

I = ΔP = P₁ - P₀ 

From its definition, the impulse has a unit of N.s, however, based on Equation (4.4), the unit of impulse is the same with the unit of momentum, since impulse is a momentum change. Thus, N.s = kg.m/s

Example 4.1

1. A wagon with 15 kg of mass moves at velocity of 0,25 m/s toward a concrete wall. At the same time, a bullet with 10 g mass is fired at velocity 1.000 m/s toward the wall. Calculate : (a)  the momentum of the wagon, and (b)  the momentum of bullet.

Answer :
m_wagon = 15 kg;  v_wagon = 0,25 m/s
m_bullet = 10 g = 10 x 10^-3 kg;   v_bullet = 1.000 m/s

(a) the momentum of the wagon

P_wagon = m_wagon  v_wagon
           = (15 kg) (0,25 m/s)
           = 3,75 kg.m/s

(b) the momentum of the bullet

P_wagon = m_wagon  v_wagon
           = (10^-2 kg) (1,000 m/s)
           = 10 kg.m/s


2.  A circus player who has 70 kg of mass is falling freely from the height of 3 m. (a) Calculate the impulse experienced by the circus player. (b) Determine the average force given by his leg when he is landing with the bending knee so that distance covered, which is measured from the first time he is touching the ground until completely stop, is 50 cm.

Exercise 4.1

1. A ball is initially at rest. The ball is then hit by a stick so that it is moving with the velocity of 10 m/s. If the mass of the ball is 250 g; calculate the impulse applied to the ball
2. A force of 70 N is applied to a body for 0.8 seconds. Determine the impulse experienced by the body.
3. A horizontal force of 200 N is working on a body, which mass is 2 kg. If the body is initially at rest and the force is working for 0.4 s, what is the final velocity of the body?
4. A stone with a mass of 500 g is thrown up vertically upward (g = 10 m/s² ). At the highest point, the stone is breaking up into two pieces with the mass ratio 2:1. If the two pieces are moving n opposite directions, determine the ratio of their kinetic energies.

Multiple Choice

1)  Impulse is classified into.......... quantity

a.  vector

b.  scalar

c.  derived and scalar

d.  fundamental

e.  vector and derived

2) The dimension of impulse is ......

a.   MLT

b.   MLT^-1

c.   ML²T

d.   MLT^-2

e.   ML²T^-2

3) A body that has a mass of 20 kg is hit by a force of 60 N. If the force is working for 0.2 s, the impulse working on the body is .............
a. 9 Ns
b. 10 Ns
c. 11 Ns
d. 12 Ns
e. 13 NS

4) A marble with mass of 200 g is sliding on a slipery floor with velocity of 20 m/s. The magnitude of momentum of the marble is ............
a. 2 kg.m/s
b. 3 kg.m/s
c. 4 kg.m/s
d. 5 kg.m/s
e 6 kg.m/s

5) An apple with mass of 200 g is falling down from a 5 m high tree, (g = 10 m/s² ). The momentum of the apple when it hits the ground is ............
a. 12 kg.m/s
b. 10 kg.m/s
c. 9 kg.m/s
d. 8 kg.m/s
e 7 kg.m/s

6)   A bullet with a mass of 8 gram is fired to a block with a mass of 250 gram, which is initially at rest on the edge of a table with the height of 1 meter. The bullet is buried into the block and after the collision, the block landed on the floor at a distance of 10-meter measure form the legs of the table. The initial velocity of the bullet is......
A)  224 m/s
B)  184 m/s
C)  145 m/s
D)  42 m/s
E)  86 m/s

7)  In a collision process, the following statement that always holds is ...........
A)  the law of momentum conservation
B)  The law of kinetic energy conservation
C)  the law of mechanical energy conservation 

D)  the law of momentum conservation and the law kinetic energy conservationE)  the law of momentum conservation and the low of mechanical energy conservation.

8)  A body with the mass of 2.5 kg is falling freely from the height of 3 meters above the floor (g = 10 m/s² ). If the body undergoes a completely inelastic collision with the floor, then the heat produced is ...
A)  7.5 cal
B)  18 cal
C)  30 kcal
D)  75 cal
E)  300 cal

9)  A body with mass of 0.1 kg is moving to the west with velocity of 0.2 m/s, and having a completely elastic collision with another body with mass of 0.15 kg, which is initially at rest. The velocity of the 0.10 kg body after collision is ......
A)  0.16 m/s to the east
B)  0.16 m/s to the west
C) 0.08 m/s to the west
B) 0.04 m/s to the east
E) 0.04 m/s to the west

10)  Two balls with masses of m1 = 20 kg and m2 = 25 kg respectively are moving into the same direction on a straight line. Ball 1 is moving with velocity of 8 m/s situated behind ball 2, which is moving with velocity of 4 m/s. If both balls are colliding inelastically with e = 0.3, the magnitude of velocities of the two balls respectively are....
a.  6.31 m/s and 5.11 m/s
b.  6.41 m/s and 5.11 m/s
c.  6.31 m/s and 5.31 m/s
d.  6.51 m/s and 5.11 m/s
e.  6.31 m/s and 5.41 m/s

Momentum momentum and impluse physics

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