Chapter 6: Momentum And Collisions. The linear momentum p of an object is the product of the object s mass m and velocity v

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1 Chapter 6: Momentum And Collisions The linear momentum p o an object is the product o the object s mass m and velocity v r r p = mv Linear momentum is a vector quantity and has the same direction as the velocity. SI Unit o Momentum: kg m / s or N s When a single, constant orce acts on the object, there is an impulse delivered to the object I r r r I = FΔt is deined as the impulse. It s a vector quantity, the direction is the same as the direction o the orce Average orce in impulse The impulse o a orce is the product o the average orce F and the time interval Δt during which the orce acts: impulse = F Δt

2 Impulse-Momentum Theorem F Δt = (m a) Δt = m (a Δt ) = m Δv= Δ(mv) IMPULSE MOMENTUM THEOREM When a net orce acts on an object, the impulse o the net orce is equal to the change in momentum o the object. Impluse = Change in momentum. F Δ t = r mv r m v i Example A 3.00-kg steel ball strikes a massive wall at 0.0 m/s at an angle o 60.0 with the plane o the wall. It bounces o the wall with the same speed and angle. I the ball is in contact with the wall or 0.00 s, what is the average orce exerted by the wall on the ball?

3 Impulse Applied to Auto Collisions The most important actor is the collision time, the time it takes the person to come to a rest Smaller momentum change and longer impact time reduce the chance o dying in a car crash Ways to increase the time Air bags F Δ t = r mv r m v i Conservation o Momentum The total momentum o an isolated system is conserved. m v r + m v r = m v r + m v r i i A result o Newton s Third Law. An isolated system is a system where the sum o all external orces is zero. 3

4 Conserv. O Momentum (not necessarily KE) total KE decreases total KE increases Momentum is conserved in either case. Conservation o Momentum Momentum is a vector quantity Direction is important Be sure to have the correct signs Remember conservation o momentum applies to the system You must deine the isolated system Example: Three carts o masses 4.0 kg, 0 kg, and 3.0 kg move on a rictionless horizontal track with speeds o 5.0 m/s, 3.0 m/s, and 4.0 m/s. The carts stick together ater colliding. Find the inal velocity o the three carts. 4

5 Elastic and Inelastic Collisions Elastic collision -- One in which the total kinetic energy o the system ater the collision is equal to the total kinetic energy beore the collision Inelastic collision -- One in which the total kinetic energy o the system is not the same beore and ater the collision. Perect D Inelas. Coll. m v + m v mv i v = m + + = m v m v m i i + m v i Collision in One Dimension (Elastic) For any collision, momentum is conserved. We have m v m v + m v = m v + 0 i For elastic coll., KE is conserved. i + mv = mv + 0 Solving or v and v, we get m m = m v = v i v v i m + m m m + Perect D Elastic Coll. 5

6 Collisions In Two Dimensions Momentum is always conserved. m v x + mv x = mv ix + m v ix m v y + mv y = mv iy + m v iy I kinetic energy is also conserved (i.e. elastic collision), mv + mv = mv i + m v i Solve one dimensional elastic collison Rocket Propulsion Cars, boats, airplanes accelerate by pushing against something (external). Rocket in space operates by discharging part o itsel at high speed. The rocket is accelerated as a result o the thrust o the exhaust gases. This represents the inverse o an inelastic collision Momentum is conserved Kinetic Energy is increased (at the expense o the stored energy o the rocket uel) 6

7 Rocket Propulsion, initial state The initial mass o the rocket is M + m M is the mass o the rocket m is the mass o the uel The initial velocity o the rocket is The speed o the uel is v e relative to the rocket Rocket Propulsion, inal state The rocket s mass is M The mass o the uel, m, has been ejected, with speed v-v e relative to the Earth The rocket s speed has increased to 7

8 Rocket Propulsion, momentum conservation ( M + Δm ) v = M (v + Δv ) + Δm ( v - v e ) M Δv = v e Δm And Δm=- ΔM, then: M Δv = -v e ΔM v -v i = v e ln ( M i / M ) Problem Solving or Collisions Coordinates: Set up coordinate axes and deine your velocities with respect to these axes It is convenient to choose the x- or y- axis to coincide with one o the initial velocities Draw: In your sketch, draw and label all the velocities and masses 8

9 Problem Solving or Collisions, Conservation o Momentum: Write expressions or the x and y components o the momentum o each object beore and ater the collision Write expressions or the total momentum beore and ater the collision in the x-direction and in the y-direction Conservation o Energy: I the collision is elastic, write an expression or the total energy beore and ater the collision Problem Solving or Collisions, 3 Solve or the unknown quantities Solve the equations simultaneously There will be two equations or inelastic collisions There will be three equations or elastic collisions 9

10 Example Problems 30. An 8.00-g bullet is ired into a 50-g block that is initially at rest at the edge o a table o height.00 m (Fig. P6.30). The bullet remains in the block, and ater the impact the block lands.00 m rom the bottom o the table. Determine the initial speed o the bullet. 57. A kg block is released rom rest at the top o a rictionless track.50 m above the top o a table. It then collides elastically with a.00-kg block that is initially at rest on the table, as shown in Figure P6.57. (a) Determine the velocities o the two blocks just ater the collision. (b) How high up the track does the kg block travel back ater the collision? (c) How ar away rom the bottom o the table does the.00-kg block land, given that the table is.00 m high? (d) How ar away rom the bottom o the table does the kg block eventually land? More Problems 6. A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with orce constant k =.00 x 0 4 N/m, as in Figure P6.6. The cannon ires a 00-kg projectile at a velocity o 5 m/s directed 45.0 above the horizontal. (a) I the mass o the cannon and its carriage is kg, ind the recoil speed o the cannon. (b) Determine the maximum extension o the spring. (c) Find the maximum orce the spring exerts on the carriage. (d) Consider the system consisting o the cannon, carriage, and shell. Is the momentum o this system conserved during the iring? Why or why not? 65. A block o mass m lying on a rough horizontal surace is given an initial velocity o v 0. Ater traveling a distance d, it makes a head-on elastic collision with a block o mass m. How ar does the second block move beore coming to rest? (Assume the coeicient o riction μ k is the same or both blocks.) 0

11 Summary o Chapter 6 Impulse is product o orce and duration Linear momentum is deined as product o mass and velocity Impulse-momentum theorem relates the two. Linear momentum or a closed system is conserved. Elastic and inelastic collisions. Rocket propulsion.