CHAPTER 3: Momentum and Impulse (2 Hours)

CHAPTER 3: Momentum and Impulse(2 Hours)

Learning Outcome: 3.1 Momentum and Impulse (1 hour) At the end of this chapter, students should be able to: • Define momentum. • Define impulse J = Ft and use F-tgraph to determine impulse. • Use

3.1 Momentum and Impulse Momentum • is defined as the product between mass and velocity. • is a vector quantity. • Equation : • The S.I. unit of linear momentum is kg m s-1. • The direction of the momentum is the same as the direction of the velocity. Momentum can be resolve into vertical (y) component & horizontal (x) component.

Impulse, • Let a single constant force, F acts on an object in a short time interval (collision), thus the Newton’s 2nd law can be written as • is defined as the product of a force, Fand the time, t OR the change of momentum. • is a vector quantity whose direction is the same as the constant force on the object. where

The S.I. unit of impulse is N s or kg m s1. • If the force acts on the object is not constant then • Since impulse and momentum are both vector quantities, then it is often easiest to use them in component form : where consider 2-D collision only

Shaded area under the Ft graph = impulse • When two objects in collision, the impulsive force, Fagainst time, t graph is given by the Figure 3.1. Figure 3.1

Example 3.1 A car of mass 800 kg is travelling at 25 m/s. Find the constant force needed to stop it in 4 seconds. Solution

1 1 Wall (2) Example 3.2 : A 0.20 kg tennis ball strikes the wall horizontally with a speed of 100 ms1 and it bounces off with a speed of 70 m s1 in the opposite direction. a. Calculate the magnitude of impulse delivered to the ball by the wall, b. If the ball is in contact with the wall for 10 ms, determine the magnitude of average force exerted by the wall on the ball. Solution :

Solution : a. From the equation of impulse that the force is constant, Therefore the magnitude of the impulse is 34 N s. b. Given the contact time,

Figure 3.2 Exercise 3.1 : An estimated force-time curve for a tennis ball of mass 60.0 g struck by a racket is shown in Figure 3.2. Determine a. the impulse delivered to the ball, b. the speed of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

Learning Outcome: 3.2 Conservation of linear momentum and impulse (1 hour) At the end of this chapter, students should be able to: • State the principle of conservation of linear momentum. • State the conditions for elastic and inelastic collisions. • Apply the principle of conservation of momentum in elastic and inelastic collisions.

3.2 Principle of conservation of linear momentum • states “In an isolated (closed) system, the total momentum of that system is constant.” OR “When the net external force on a system is zero, the total momentum of that system is constant.” • In a Closed system, From the Newton’s second law, thus

then Therefore • According to the principle of conservation of linear momentum, we obtain OR The total of initial momentum = the total of final momentum

2 2 2 1 1 1 Elastic collision • is defined as one in which the total kinetic energy (as well as total momentum) of the system is the same before and after the collision. • Figure 3.4 shows the head-on collision of two billiard balls. Before collision At collision After collision Figure 3.3

The properties of elastic collision are a. The coefficient of restitution, e = 1 b. The total momentum is conserved. c. The total kinetic energy is conserved. OR

2 2 2 1 1 1 Before collision At collision After collision (stick together) Inelastic (non-elastic) collision • is defined as one in which the total kinetic energy of the system is not the same before and after the collision (even though the total momentum of the system is conserved). • Figure 3.4 shows the model of a completely inelastic collision of two billiard balls. Figure 3.4

Caution: • Not all the inelastic collision is stick together. • In fact, inelastic collisions include many situations in which the bodies do not stick. • The properties of inelastic collision are a. The coefficient of restitution, 0  e < 1 b. The total momentum is conserved. c. The total kinetic energy is not conserved because some of the energy is converted to internal energy and some of it is transferred away by means of sound or heat. But the total energy is conserved. OR

Elastic versus inelastic collision

A B Figure 3.5 Linear momentum in one dimension collision Example 3.3 : Figure 3.5 shows an object A of mass 200 g collides head-on with object B of mass 100 g. After the collision, B moves at a speed of 2 m s-1 to the left. Determine the velocity of A after Collision. Solution :

m1 m2 m1 Before collision After collision Figure 3.6 Linear momentum in two dimension collision Example 3.4 : A tennis ball of mass m1 moving with initial velocity u1 collides with a soccer ball of mass m2 initially at rest. After the collision, the tennis ball is deflected 50 from its initial direction with a velocity v1 as shown in figure 3.6. Suppose that m1 = 250 g, m2 = 900 g, u1 = 20 m s1 and v1 = 4 m s1. Calculate the magnitude and direction of soccer ball after the collision.

Solution : From the principle of conservation of linear momentum, The x-component of linear momentum,

Solution : The y-component of linear momentum, Magnitude of the soccer ball, Direction of the soccer ball,

Exercise 3.2 : An object P of mass 4 kg moving with a velocity 4 ms1 collides elastically with another object Q of mass 2 kg moving with a velocity 3 ms1 towards it. a. Determine the total momentum before collision. b. If P immediately stop after the collision, calculate the final velocity of Q. c. If the two objects stick together after the collision, calculate the final velocity of both objects. ANS. : 10 kg ms1; 5 ms1 to the right; 1.7 m s1 to the right

Figure 3.7 Exercise 3.2 : 2. A ball moving with a speed of 17 m s1 strikes an identical ball that is initially at rest. After the collision, the incoming ball has been deviated by 45 from its original direction, and the struck ball moves off at 30 from the original direction as shown in Figure 3.17. Calculate the speed of each ball after the collision. ANS. : 8.80 m s 1; 12.4 m s1

THE END… Next Chapter… CHAPTER 4 : FORCE

CHAPTER 3: Momentum and Impulse (2 Hours)