Linear Momentum AP Physics C Mrs Coyle Part

Linear Momentum AP Physics C Mrs. Coyle

Part I • Momentum • Newton’s 2 nd law expressed in terms of Momentum • Impulse • Conservation of Momentum in one Dimension

Linear Momentum, p p= mv • vector quantity • in the direction of v • unit kg m/s

Relating Force and Momentum For a particle of constant mass:

Newton’s Second Law • The time rate of change of the linear momentum of a particle is equal to the net force. • This is the form in which Newton presented the Second Law. F = dp/dt

• For a constant force: ΣF= Δp / Δt

Impulse, I = Dp (Change in Momentum) For a constant Force : Dp=m. Dv=Ft • When considering change in momentum you must consider the vector change in momentum. • If momenta are in one dimension it is enough to assign the proper sign. • If momenta are in two dimensions the momentum change must be found by subtracting the final –initial momentum vectors.

Impulse for a constant force Dp = mvf –mvi Dp= m. Dv Dp= ΣF t

Impulse • From Newton’s Second Law: F = dp/dt dp = Fdt

Change in Momentum in Two Dimensions • A tennis ball bounces as shown. Express the change in momentum in terms of m, v, q. q v Wall v Ball q

Graph • Impulse is the area under a F vs t graph. F t

Conservation on Momentum • In the absence of an external force the momentum of a system is conserved. • That means that the vector sum of the momenta before and after an impact must be equal.

Conservation of Momentum in One Dimension m 1 v 1 + m 2 v 2 = m 1 v 1 + m 2 v 2

Part II • Elastic and Inelastic Collisions • Conservation of Momentum in two Dimensions

Elastic and Inelastic Collisions • Elastic: Momentum and Kinetic Energy are both conserved. • Inelastic: Momentum is conserved but Kinetic Energy not conserved. – Perfectly Inelastic: the objects stick together during the collision.

Note • Momentum is conserved in all collisions.

Inelastic Collisions • Kinetic Energy is not conserved. • The objects are often deformed, or become attached to one another during the collision(perfectly inelastic). • Kinetic energy turns to heat, sound, etc.

Collisions in two Dimensions • For problems with collision in two dimensions, since momentum is a vector, use conservation of momentum separately in the x axis and in the y axis.

Example #34 • A proton moving with a velocity of vi I, collides elastically with another proton that is initially at rest. If the two protons have equal speeds after the collision, find: a) The speed of each proton after the collision in terms of vi b) The direction of the velocity vectors after the collision. Ans: a) v=vi / 21/2 b) 450 above +x, 450 below +x

Example #21 • A 45. 0 kg girl is standing on a 150 kg plank on a frozen lake. Initially they are both at rest. The girl then walk along the plank with a speed of 1. 50 m/s relative to the plank. Find the speed of the girl and the plank relative to the ground. Ans: vp = -0. 346 m/s, vg =1. 15 m/s

Example #25 • A 12. 0 g wad of sticky clay is hurled horizontally at a 100 g wooden block. After impact, the block slides 7. 50 m before coming to rest. If the coefficient of friction between the block and the surface is 0. 650, what was the speed of the clay immediately before impact? • Answer: 91. 2 m/s

Example #67 • A 5. 00 g bullet moving with an initial speed of 400 m/s is fired into and passes through a 1. 00 kg block. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring with a force constant of 900 N/m. If the block moves 5. 00 cm to the right after the impact find: a) The speed at which the bullet emerges from the block b) The mechanical energy converted into internal energy in the collision. Ans: a) 100 m/s, b)374 J

Example: Conservation of Momentum in Two Dimensions A firecracker traveling with a velocity of 3 m/s in the direction of the +x axis, explodes and breaks up into two equal pieces. After the explosion, one piece flies off with a velocity of 2 m/s at an angle of 30 degrees above the +x axis. Find the velocity of the second piece. Ans: Vx= 4. 27 m/s Vy=-1 m/s V=4. 39 m/s 13 degr below +x-axis

Ex: Ballistic Pendulum

Ex: Ballistic Pendulum The ballistic pendulum was used to measure speeds of bullets in the old days. The large block of wood has a mass of 5. 4 kg. A bullet of mass 9. 5 g is fired into the block. The bullet is embedded into the block and the two rise to a maximum height of 6. 3 cm. What is the speed of the bullet prior to collision? Ans: 630 m/s