JPJC 2026 Collision Lecture Notes Student

2026/JPJC/PHYSICS/9478 1 JURONG PIONEER JUNIOR COLLEGE 9478 H2 PHYSICS COLLISIONS ______________________________________________________________________ Content (1) Impulse (2) Conservation of momentum and energy Learning Outcomes Students 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.

2026/JPJC/PHYSICS/9478 2 Impulse Objective: a) recall the impulse is given by the area under the force-time graph for a body and use this to solve problem. This learning outcome was covered in the previous topic “Motion and Forces”. Recall: Newton's second law states that: The rate of change of momentum of a body is (directly) proportional to the resultant force acting on the body and is in the same direction as the resultant force. where the momentum p of a body is the product of its mass m and instantaneous velocity v. Mathematically: p = m v Momentum is a vector quantity and its unit is kg m s-1 or N s. Recall Newton’s second law: Integrating both sides give: dpF dt dp F dt dp F dt     p F dt  If a force-time ( F–t) graph is available, the change in momentum of a body, p, could be obtained from the net area under the graph . Suppose the variation of an applied force F(t) with time t is as shown in the given graph. The change in momentum, p, due to F(t) from time t1 to t2 is then given by: p = p2 – p1 = area under F–t graph where p1, p2 = momentum at t1 , t2 respectively Alternatively, if F is the average force acting, the change in momentum can also be given by the area of the rectangle F t , which has the same value as the area below the curve. This area is also known as the impulse. Impulse is the product of the average net force and the time it acts on a body. Note: Impulse of the force acting on a body can also be seen as the change in momentum of the body caused by the force. Its unit is N s or kg m s1. When you hit a tennis ball with your racket the ball changes its momentum. We say that the racket has delivered an impulse, I to the ball. p F t1 t2 t

2026/JPJC/PHYSICS/9478 3 Example 1 A body of mass 2.0 kg initially at rest is acted on by a force F which varies with time t as shown below. Determine the momentum of the body at time s. Solution The area under the F−t graph is the change in momentum experienced by the body. Example 2 To stop a car of mass 1500 kg travelling at 30 m s 1, the driver applies his brakes so that F, the total stopping force increases steadily to a maximum and then decreases to zero as shown on the graph. Determine the following quantities: a) The momentum of the car when it is travelling at 30 m s 1, b) the change in momentum between t = 0 and t = 10 s, c) the value of Fmax, and d) the value of Faverage Solution 6t p F / N t / s 8 0 1 2 3 4 5 6 0 t /s 0 Fmax F / N 20

2026/JPJC/PHYSICS/9478 4 Conservation of momentum and energy Objectives: (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.  Applying Newton’s third law to closed collision systems will lead us to the principle of conservation of momentum.  By a system, we mean a set of chosen objects which may interact with each other. A closed (or isolated) system is one in which the only (significant) forces are those between the objects in the system. Typically, we consider a closed system of two interacting bodies.  Consider an object 1, of mass m1 and velocity u1, colliding with object 2, of mass m2 and velocity u2, moving in the same direction. (u1 will have to be larger than u2 in order for them to collide.)  During the collision, an average force F is exerted by object 1 on object 2 (right). By Newton’s third law, an equal and oppositeF is exerted by object 2 on object 1 (left).  We consider average force F in this case because the force may not be constant during the duration of contact.  In the very short time interval t that both objects are in contact with each other, they will experience the same magnitude of impulse F t but opposite in direction.  If object 1 moves with a reduced velocity v1 after collision, object 2 will move with an increased velocity v2 : Taking right to be the positive direction: m1 m2 1 2 u2 u1 Before collision m1 m2 1 2 v2 v1 After collision 1 2 During collision

2026/JPJC/PHYSICS/9478 5 Change in momentum of object 1: 1 1 1 1 1F t p m v m u     -------------- (1) Change in momentum of object 2: 2 2 2 2 2F t p m v m u    -------------- (2) NOTE: 1p is negative; 2p is positive. Substituting (2) into (1) gives: 1 1 1 1 2 2 2 2 ( )m v m u m v m u   1 1 1 1 2 2 2 2m v m u m v m u    Therefore 1 1 2 2 1 1 2 2m u m u m v m v   Total momentum of the system = Total momentum of the system before collision after collision  The principle of conservation of momentum states that: The total momentum of a system is constant, provided no external resultant force acts on the system.  The principle of conservation of momentum can also be understood using the following equation: 1 2 0p p    The principle of conservation of momentum is applicable to any system as long as there is no net external force acting on the system.  The principle of conservation of momentum was proven by assuming that the directions of travel of objects 1 and 2 are the same before and after the collision, but the results are valid even if they were travelling in opposite directions.  There are two common closed systems within the scope of this syllabus: - collisions and - disintegration. Collisions There are two types of collisions: elastic collisions and inelastic collisions. (Perfectly) Elastic Collisions  (Perfectly) elastic collisions are those in which the total kinetic energy is conserved.  Truly elastic collisions can only occur in practice on an atomic scale i.e. collisions between atoms and molecules.  By principle of conservation of momentum: 1 1 2 2 1 1 2 2m u m u m v m v   ---- (1)  By conservation of K.E.: K.E. of system before collision = K.E. of system after collision

2026/JPJC/PHYSICS/9478 6 2 2 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 2 2m u m u m v m v   ---- (2) These two equations can be used to solve problems involving elastic collisions. From equations (1) and (2),