1. 1 SACE Stage 2 Physics Momentum in 2-Dimensions

2. 2 Vector Form of Newton’s Second Law of Motion Consider a particle reflecting off a surface without a change in speed. A particle has m = 2 kg and initial velocity v i = 20 m s -1 and the collision lasts for  t = 0.2 s, then determine the average force acting on the particle. vivi vfvf m v f = -20 ms -1 Acceleration a is given by,

3. 3 Momentum in 2-Dimensions Vector Form of Newton’s Second Law of Motion vivi vfvf m a = -200ms -1 Acceleration is in opposite direction to v i. and since F = m a then F = 2 x (-200) = -400 N ie. 400 N away from the wall Ie, the wall applied a force of 400N on the ball away from the wall and due to Newtons 3 rd Law, the ball exerted a force of 400N towards the wall.

4. 4 Momentum in 2-Dimensions Vector Form of Newton’s Second Law of Motion vivi vfvf m For a particle not directed at 90 o to the surface: The acceleration does not act in the direction of the vertical component therefore the vertical component remains constant and the horizontal component will change.

5. 5 Momentum in 2-Dimensions Newton’s Second Law of Motion in Terms of Momentum Momentum has units of kg.m.s -1, is a vector quantity and is defined has, Newtons 2 nd Law is : and Hence,

6. 6 Momentum in 2-Dimensions Newton’s Second Law of Motion in Terms of Momentum So Newton’s second law becomes, i.e. But p=mv F and p are in the same direction and are vectors The force acting on an object can now be described in terms of how quickly its momentum is changing.

7. 7 Momentum in 2-Dimensions The Law of Conservation of Momentum Consider two objects in a region free from all other forces: If two objects collide then the first object applies a force of F 1 on the second object and due to Newtons 3 rd Law, the second object exerts a force of F 2 on the first object. By Newton’s second law then:

8. 8 Momentum in 2-Dimensions The Law of Conservation of Momentum Since object 1 would be in contact with object 2 for the same amount of time object 2 is in contact with object 1 then  t 1 =  t 2  Thus when no net external forces are applied to a system of two bodies, the total momentum of the system before is the same as the total momentum of the system after.

9. 9 Momentum in 2-Dimensions The Law of Conservation of Momentum Law of conservation of linear momentum If there are no net external forces acting on a system of bodies then the total linear momentum of the system is conserved. NB.(1) No net external forces means an isolated system. (2) Total linear momentum refers to both size and direction (vector addition is needed to find the total) (3) Conserved means that it stays the same (throughout the interaction) (4) The objects may be deformed, broken apart, stuck together or exploded. However momentum is still conserved as long as no net external forces act.

10. 10 Momentum in 2-Dimensions The Law of Conservation of Momentum Example Consider two objects interacting along a straight line, where friction is negligible.

11. 11 Momentum in 2-Dimensions The Law of Conservation of Momentum Example A rail wagon of mass 4.0 tonnes, moving 3ms -1 North collides with another wagon of mass 2.0 tonnes. They stick together. What is their combined velocity?

12. 12 Example A rail wagon of mass 4.0 tones, moving 3ms -1 North collides with another wagon of mass 2.0 tones. They stick together. What is their combined velocity? Take North to be the positive direction, Momentum in 2-Dimensions The Law of Conservation of Momentum

13. 13 Momentum in 2-Dimensions The Law of Conservation of Momentum Assuming the system isolated, linear momentum is conserved.

14. 14 Momentum in 2-Dimensions The Law of Conservation of Momentum Example A body at rest explodes breaking into 3 pieces. One piece flies off due East at 30ms -1. Another piece flies of due North at 30ms -1. What is the velocity of the 3 rd piece if its mass is 3 times that of the other pieces?

15. 15 Momentum in 2-Dimensions The Law of Conservation of Momentum Example A body at rest explodes breaking into 3 pieces. One piece flies off due East at 30ms -1. Another piece flies of due North at 30ms -1. What is the velocity of the 3 rd piece if its mass is 3 times that of the other pieces? Assuming the system is isolated, linear momentum is conserved.

16. 16 Momentum in 2-Dimensions Collisions involving 2-Dimensions Consider the following multi image photograph of two pucks colliding on an air table. Time taken between frames is 1s and consider image to be actual size. Refer to hand out 1 m 1 = 3 kg m 2 = 2 kg

17. 17 Momentum in 2-Dimensions Rockets Consider a stationary rocket in outer space to be an isolated system. Total Momentum = 0

18. 18 Momentum in 2-Dimensions Rockets Total Momentum = 0 Momentum of exhaust gases Momentum of Rocket The rocket is propelled forward and momentum is conserved (Assume time for taken for fuel to burn is very small so that the mass of fuel can be considered to be constant).

19. 19 Momentum in 2-Dimensions Rockets Changing Direction in Outer Space The space craft must orientate itself at right angles so that the force provided by the fuel is at right angles to the motion of the rocket. The rocket must continually change it orientation to ensure that the force provided by the fuel is always directed to the centre of the circular arc. This thrust will provide the rocket with the centripetal acceleration needed.

20. 20 Momentum in 2-Dimensions Rockets Some rockets also propel ions out of ion thrusters achieving he same as propelling a gas. Ion thrusters have less acceleration but need to carry much less fuel. Significant velocities are achieved very long periods of time. Solar Sails Solar sails can accelerate a craft by reflecting or absorbing photons from the sun. If a single photon of x-units of momentum is ABSORBED it will give the craft x-units of momentum, because it loses all of its momentum to the craft (LOCM).

21. 21 Momentum in 2-Dimensions Rockets Solar Sails If the sail REFLECTS a photon of x-units of momentum the craft will gain 2x units of momentum because the photon changes 2x units:  craft gains 2x→→ to conserve total p.