Momentum

1. Momentum Physicists have defined aquantity called momentum(動量) of amoving object as... Momentum = mass velocity P = mv Unit: kg m s1 A vector quantity

2. FBA FAB Newton’s 3rd law and Conservation of momentum • Newton’s 3rd law • FBA = -FAB A B If there is no external force acting on a system, then the total momentum of the system is conserved

3. 100 ms-1 h v Example 1A bullet of mass 10 g traveling horizontally at a speed of 100 ms-1 embeds itself in a block of wood of mass 990 g suspended by strings so that it can swing freely. Find(a) the vertical height through which the block rises, and(b) how much of the bullet’s energy becomes internal energy. • Solution: (a) Let v be the velocity of the block just after impact. By conservation of momentum, (0.01)(100) = (0.01 + 0.99)v v = 1 ms-1 By conservation of energy, ½ mv2 = mgh ½ (1)2 = 10h h = 0.05 m

4. 100 ms-1 h v Example 1A bullet of mass 10 g traveling horizontally at a speed of 100 ms-1 embeds itself in a block of wood of mass 990 g suspended by strings so that it can swing freely. Find(a) the vertical height through which the block rises, and(b) how much of the bullet’s energy becomes internal energy. • Solution: (b) Some of the K.E. is converted into internal energy. Energy required = ½ mu2 – ½ (m + M)v2 = ½ (0.01)(100)2 – ½ (0.01 + 0.99)(1)2 = 49.5 J

5. Application 1 – measuring the inertial mass • Weighing machine or beam balance only measures the weight. • To find the mass (gravitational mass), we must depend on the equation m = W/g. • However, g varies from place to place and we cannot use this method to find the mass in the outer space. • Another way to determine the mass (inertia mass) without depending on the value of g is to use the principle of conservation of momentum.

6. v1 v2 m1 m2 m1 m2 Before explosion After explosion Application 1 – measuring the inertial mass • Consider two trolleys of m1 and m2 are in contact. A spring is used to cause them to explode, moving off the velocity v1 and v2. • If no external force acts on them, conservation of momentum gives 0 = m1v1 + m2v2 • Therefore, by measuring their velocities, the ratio of their masses can be found. • If a standard trolley is used, the other mass can be found.

7. 0.16 ms-1 0.96 ms-1 m1 m2 m1 m2 Before explosion After explosion Example 2Compare the masses of 2 objects X and Y which are all initially at rest. They explode apart and their speeds become 0.16ms-1 and 0.96ms-1 respectively. • By conservation of momentum, 0 = m1(-0.16) + m2 (0.96) • The ratio of the masses m1: m2 = 6:1

8. Applications 2 – Rocket engine • The rocket engine pushes out large masses of hot gas. • The hot gas is produced by mixing the fuel (liquid hydrogen) with liquid oxygen in the combustion chamber and burning the mixture fiercely. • The thrust arises from the large increase in momentum of the exhaust gases.

9. Applications 2 – Jet engine • A jet engine uses the surrounding air for its oxygen supply. • The compressor draws in air at the front, compresses it, fuel is injected and the mixture burns to produce hot exhaust gases which escape at high speed from the rear of the engine. • These cause forward propulsion.

10. gas bullet rifle Recoil of rifles Two factors affects the recoil speed of a rifle. 1 momentum given to the bullet 2 momentum given to the gases produced by the explosion.

11. Example 3A rife fires a bullet with velocity 900 ms-1 and the mass of the bullet is 0.012 kg. The mass of the rifle is 4 kg and the momentum of gas ejected is about 4 kg ms-1. Find the velocity of the recoil of the rifle. • Solution: By conservation of momentum 0 = 0.012 x 900 + (-4) + 4v v = -1.7 ms-1 The recoil velocity is 1.7 ms-1 backward 900 ms-1 gas (4 kg ms-1) v rifle bullet

12. mvmu F = t Rearrange terms in  Ft=mv  mu impulse:productof force & time duringwhich the force acts (vector) Impulse = change in momentum

13. a Force-time graph of impact force / N force / N time / s time / s Area under F-t graph = impulse = change in momentum

14. Collisions General properties • A large force acts on each colliding particles for a very relatively short time. • The total momentum is conserved during a collision if there is no external force. • In practice, if the time of collision is small enough, we can ignore the external force and assume momentum conservation. • For example, when a racket strikes a tennis ball, the effect due to gravitational force (external force) is neglected since the time of impact is very short.

15. Collisions • For another example, when a cannon fires a metal ball, the effect due to frictional force (external force) on the cannon is neglected since the time of explosion is very short.

16. Different kinds of collisions • Assume no external force acts on colliding bodies. Yes Yes Yes No No (K.E. loss is maximum) Yes

17. u2 v2 u1 v1 Before collision After collision Relative velocity rule (For elastic collisions only) • relative speed before collision = relative speed after collision u1 – u2 = v2 – v1

18. Find the velocities of A and B after the elastic collision. 3 ms-1 v2 5 ms-1 v1 • By conservation of momentum, (1)(5) + (2)(3) = (1)(v1) + (2)(v2) v1 + 2v2 = 11 --- (1) • By relative velocity rule, 5 – 3 = -(v1 – v2) v2 – v1 = 2 --- (2) • ∴ v1 = 2.33 ms-1 and v2 = 4.33 ms-1 1 kg 2 kg Before collision After collision

19. v V At rest u A B A B Trolley A of mass m with velocity u collides elastically with trolley B of mass M at rest. Find their velocities v and V after collision. • By conservation of momentum, mu = mv + MV --- (1) • By relative velocity rule, u = V – v --- (2) • (1) + (2) x m: 2mu = (m + M)V ⇒ • (1) – (2) x M: mu – Mu = (m + M)v ⇒

20. Trolley A of mass m with velocity u collides elastically with trolley B of mass M at rest. Find their velocities v and V after collision. Some interesting results • If m = M, trolley A will ___________________________ • If m < M, trolley A will ________________________ • If m << M, trolley B will __________________________ At rest u A B A B stop after the collision move in the opposite direction after the collision remain at rest after the collision

21. y Mv2 M m a mu x b mv1 Collisions in two dimensions • Momentum can be resolved along different directions. • Consider the following oblique impact. A particle of mass m with velocity u collides with another stationary particle of mass M obliquely as shown below. Apply conservation of momentum along x-axis, mu = mv1cos b + Mv2cos a Apply conservation of momentum along y-axis, 0 = Mv2sin a + (-mv1sin b)

22. 500 g Before explosion A stationary object of mass 500 g explodes into three fragments as shown below. Find the speed the two larger fragments v1 and v2 after the explosion. 0.2v2 200 g • Along vertical direction: 0 = 0.2 v2 sin 30o – 0.1(40) sin 45o v2 = 28.3 ms-1 • Along horizontal direction: 0 = 0.2v2cos30o + 0.1(40)cos45o – 0.2v1 v1 = 38.6 ms-1 200 g 30o 0.2v1 45o After explosion 100 g 0.1 x 40 ms-1

23. Right-angled fork track • If one particle collides with another identical particle obliquely, the angle between the directions of motion of the two is always 90o if the collision is elastic. Significance • Alpha particles travel in a cloud chamber and have collisions with helium atoms. • The figure above shows after collision the alpha particle and the helium atom travel at right angle to each other. • This implies alpha particles must have the same mass as helium atom and actually they are nuclei of helium.

24. Mathematical proof of right-angled fork track (i.e. q = 90o) u a v2 a collision between identical particles q v1 Along the line of centres, Perpendicular to the line of centres, For elastic collision,

25. (1)2 + (2)2: Sub.(4) into (3): The two particles must move at right angle to each other.

26. Elastic collision between a smooth ball and a fixed surface • Momentum along the fixed surface is unaltered by the impact. v v Force of impact • Momentum along the fixed surface is unaltered by the impact.

27. v q f v Elastic collision between a smooth ball and a fixed surface • Since the collision is elastic, after the impact, it rebounds with the same speed. • Momentum along the fixed surface is unaltered by the impact. mv sin q = mv sin f⇒q = f ⇒ the angle of reflection = the angle of incident • Change in momentum perpendicular to the surface = mv cos q – (-mv cos f) = 2mv cos q. • Force of impact = 2mv cos q / t