RI Chap 6 Collisions - Lecture Notes

6 COLLISIONS H2 Physics 9478 Content Page 6.1 Impulse and Momentum Change 2 6.2 Conservation of Momentum and Energy 6 6.3 Collisions 6 6.4 Collisions with increase in total kinetic energy 12 6.5 Appendix 15 Learning Outcomes Candidates should be able to: (a) recall that impulse is given by the area under the force-time graph for a body and use this to solve problems. (b) state the principle of conservation of momentum. (c) apply the principle of conservation of momentum to solve simple problems including inelastic and (perfectly) elastic interactions between two bodies in one dimension (knowledge of the concept of coefficient of restitution is not required). (d) show an understanding that, for a (perfectly) elastic collision between two bodies, the relative speed of approach is equal to the relative speed of separation. (e) show an understanding that, whilst the momentum of a closed system is always conserved in interactions between bodies, some change in kinetic energy usually takes place.

Page | 2 Introduction In the topic Fields and Energy, we have seen that total energy is always conserved within an isolated system, and bodies within the system transfer energies between each other when they interact. During their interactions (such as in collisions), these bodies exert forces on each other, hence transferring momenta too. We will learn about the conservation law associated with momentum, and how it can be applied to interacting bodies. 6.1 Impulse and Momentum Change  Momentum and Newton’s Second Law of Motion Recall that the momentum p of a body is defined as the product of its mass m, and its velocity v. p mv= Newton’s Second Law of Motion states that the rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction as the resultant force. In S.I. units, the proportionality constant is 1. Hence, net dpF dt= In this topic, we will examine the effect of applying a resultant force on a body over a duration of time, and see how this concept, together with Newton’s Third Law of Motion naturally lead to the principle of conservation of momentum.  Impulse Consider a resultant force Fnet acting on a body . The graph in Fig. 6.1 shows how Fnet varies with time t. Fig. 6.1 From Newton’s Second Law, net dpF dt= . During a small time interval d t, the momentum of the body changes by netdp F dt= . t1 t2 Fnet / N Fmax t/s dt

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 3 To determine the change in momentum p∆ over a time interval t1 to t2, 2 1 2 1 net net f i pt pt t fi t dp F dt p p p F dt = − = ∆= ∫∫ ∫ where 2 1 net t t F dt∫ is the area under the netFt − graph from t1 to t2. Since the resultant force may be varying, it is sometimes necessary to define an average resultant force netF〈〉 . The average resultant force netF〈〉 over the same time interval, 21tt t∆= − , produces the same change in momentum as the varying resultant force. netpF t∆= ∆ Hence impulse of the force acting on a body over a time interval t∆ is equal to its change in momentum over the same time interval t∆ . 2 1 impulse t net nett p F t F dt= ∆= ∆= ∫ Impulse is a vector. It has the same S.I. unit as momentum. S.I. unit for impulse: 1kg m s− or N s  Concept of Impulse Have you wondered why we have to buckle our seat belts in cars? Or why cars crumple during a car crush? The crumpling of a car during a car crash is not due to the poor quality of the car frame! It is a safety feature designed to reduce the impact of the crash on the passengers in it. To reduce the injury on passengers, the average resultant force acting on them should be reduced. For a car travelling at a particular speed (which is also the passenger’s speed), the impulse on the passenger or the change in momentum of the passenger due to the car crash is constant as it reduces his / her initial speed to zero. From impulse netFt= ∆ , the average resultant force acting on the passenger can be reduced if the time taken for the crash, t∆ is increased. That is why cars crumple during the crash; it increases the duration of the crash! Similarly, the air bag also increases the time taken for the passenger ’s momentum to reduce to zero and therefore reduce the force acting on the passenger based on the Newton’s second law of motion. Impulse and Change in Momentum www.youtube.com/watch ?v=FOlrZF4WHXg Impulse Impulse is the product of the average resultant force netF〈〉 acting on the body and the time interval t∆ for which the force acts. impulse netFt= ∆

Page | 4 Example 1 (constant force) A block of mass 2.0 kg is moving with an initial velocity of 5.0 m s −1 towards the right on a smooth horizontal surface. A constant force of 10 N towards the right is then applied on it for 2.0 s. (a) Calculate the change in momentum experienced by the block. (b) Determine the final velocity of the block. (c) Determine the final velocity of the block if its initial velocity is 5.0 m s−1 towards the left. Recall that velocity is a vector quantity. Example 2 (time varying force) A body of mass 3.0 kg is moving with an initial speed of 1.0 m s−1 when it is acted upon by a force F which varies with time t as shown below: If the force acts in the same direction as the initial velocity of the body, what is the body’s velocity at t = 10.0 s? 10 N initially force applied for 2.0 s 5.0 m s−1 0 2.0 5.0 6.0 10.0 t / s F / N 6.0

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 5 Example 3 (average of a time varying force) A man throws a ball of mass 0.30 kg with a horizontal speed of 20 m s−1. (a) His hand is in contact with the ball for a time interval of 0.20 s while throwing the ball. Calculate the average force he exerts on the ball. (b) The ball thrown by the man hits a vertical wall at right angles with a speed of 20 m s -1 and bounces back horizontally with a speed of 15 m s-1. The ball is in contact with the wall for 0. 050 s. Determine the magnitude of the average force exerted by the wall on the ball. Recall that momentum is a vector quantity. Example 4 A parachutist of weight W strikes the ground with her knees bent and comes to rest with a deceleration of 3g. (a) Determine, in terms of W, the force exerted on her by the ground during landing. (b) Suggest and explain w hat may happen if the parachutist did not bend her knees upon landing? Recall the safety features of a car. N W a

Page | 6 6.2 Conservation of Momentum and Energy  Principle of Conservation of Momentum In closed system of bodies (one where no resultant external force is acting), when the bodies interact with one another, their individual momentum may change but the total momentum of the bodies remains constant. This is the principle of conservation of momentum. An external force is a force acting on the system from the outside. This principle is a consequence of Newton’s Second Law and Third Law of Motion. The principle will always hold if there is no net external force acting in the direction in which momentum is considered. Whilst the momentum of a close