Momentum and Collisions Phy 114
Momentum Momentum: p = mv Units are kg(m/s): no derived units A vector quantity: same direction as velocity v=2m/s p= 3 kg (2m/s)
From Newton s 2nd ΣF = ΣF = ma v m t ΣF( t) = m( v) ΣF( t) = mv ΣF( t) = p 2 mv 1 Impulse: J=ΣF Δt Units are same as momentum Impulse causes a change in momentum What happens to ΣF with Small Δt? Large Δt?
The Impulse-Momentum Theorem Impulse and momentum are related as: An impulse delivered to an object causes the object s momentum to change. Impulse can be written in terms of its x- and y-components: 2015 Pearson Education, Inc.
Impulse Impulse has units of N s, but N s are equivalent to kg m/s. The latter are the preferred units for impulse. The impulse is a vector quantity, pointing in the direction of the average force vector: 2015 Pearson Education, Inc.
Graphs A force is acting in the x-direction on a 4.00- kg particle. The force varies in time and is shown on the graph. Find: the impulse the final velocity if initially at rest the final velocity if initial velocity is 2.00 m/s. Force (N) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Force vs Time for a Moving Cart 0 5 10 15 20 Time (sec)
Impulse It is useful to think of the collision in terms of an average force F avg. F avg is defined as the constant force that has the same duration Δt and the same area under the force curve as the real force. 2015 Pearson Education, Inc.
Example: Abrupt stop in a car A 60-kg person is traveling in a car that is moving at 16 m/s with respect to the ground when the car hits a barrier. The person is not wearing a seat belt, but is stopped by an air bag in a time interval of 0.20 s. Determine the average force that the air bag exerts on the person while stopping him. 2014 Pearson Education, Inc.
Example: Stopping the fall of a movie stunt diver The record for the highest movie stunt fall without a parachute is 71 m, held by 80-kg A. J. Bakunas. His fall was stopped by a large air cushion, into which he sank about 4.0 m. His speed was approximately 36 m/s when he reached the top of the air cushion. Estimate the average force that the cushion exerted on this stunt diver's body while stopping him. 2014 Pearson Education, Inc.
Problem Solving Multiple Plans Draw a MD What new motion equation can you add? Draw a FBD What new equation do we have that relates force and motion? Ans: F avg = 375 N A 0.500-kg football is thrown toward the east with a speed of 15.0 m/s. A stationary receiver catches the ball and brings it to rest in 0.020 0 s. What is the average force exerted on the receiver?
Problem Solving: solve this way or Multiple Plans Draw a MD What new motion equation can you add? Draw a FBD x=0 t=0 v=15 m/s a= x= t= 0 What new equation do we have that relates force and motion? Ans: F avg = 375 N n hand on football = ma x
Problem Solving:..this way Multiple Plans Draw a MD What new motion equation can you add? Draw a FBD What new equation do we have that relates force and motion? Impulse --- Momentum equation n football t hand on = m v Ans: F avg = 375 N
Example 9.2 Calculating the change in A ball of mass m = 0.25 kg rolling to the right at 1.3 m/s strikes a wall and rebounds to the left at 1.1 m/s. What is the change in the ball s momentum? What is the impulse delivered to it by the wall? momentum 2015 Pearson Education, Inc.
Example 9.2 Calculating the change in SOLVE Thex-component of the initial momentum is momentum (cont.) The y-component of the momentum is zero. After the ball rebounds, the x-component is 2015 Pearson Education, Inc.
Example 9.2 Calculating the change in Watch the signs! If positive is to the right, then negative is to the left. momentum (cont.) ASSESS The impulse is negative, indicating that the force causing the impulse is pointing to the left, which makes sense. 2015 Pearson Education, Inc.
Collisions Elastic Momentum is Conserved Kinetic Energy is Conserved Inelastic Momentum is Conserved Kinetic Energy is NOT Conserved
Example 9.5 Speed of ice skaters pushing off Two ice skaters, Sandra and David, stand facing each other on frictionless ice. Sandra has a mass of 45 kg, David a mass of 80 kg. They then push off from each other. After the push, Sandra moves off at a speed of 2.2 m/s. What is David s speed? 2015 Pearson Education, Inc.
Example 9.7 Recoil speed of a rifle A 30 g ball is fired from a 1.2 kg spring-loaded toy rifle with a speed of 15 m/s. What is the recoil speed of the rifle? PREPARE As the ball moves down the barrel, there are complicated forces exerted on the ball and on the rifle. However, if we take the system to be the ball + rifle, these are internal forces that do not change the total momentum. 2015 Pearson Education, Inc.
Example 9.7 Recoil speed of a rifle The external forces of the rifle s and ball s weights are balanced by the external force exerted by the person holding the rifle, so (cont.) This is an isolated system and the law of conservation of momentum applies. 2015 Pearson Education, Inc.
SOLVE Example 9.7 Recoil speed of a rifle (cont.) Conservation of momentum gives 2015 Pearson Education, Inc.
Problem in 2-D Collision Momentum always conserved Do this first Remember it is a vector Kinetic Energy conserved in elastic collision Check this next A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.2 m/s at an angle of 30 with respect to the original line of motion. Find the velocity (magnitude and direction) of the second ball after collision. Was the collision inelastic or elastic?
Conservation of Momentum Motion Momentum Momentum (x) Momentum (y) Mass 1: Initial 5 m/s Mass 2: Initial 0 m/s Mass 1: Final 4.2 m/s at 30 Mass 2: Final Conserve Momentum
Conservation of Momentum Assume 0.5 kg balls Motion Momentum Momentum (x) Momentum (y) Mass 1: Initial 5 m/s Mass 2: Initial 0 m/s Mass 1: Final 4.2 m/s at 30 2.5 kgm/s to the right +2.5 kgm/s 0 0 0 0 2.1 kgm/s at 30 +2.1 cos30 = 1.82 kgm/s +2.1 sin30 = 1.05 kgm/s Mass 2: Final unknown p x p y Conserve Momentum 2.5 + 0 = 1.82 + p x P x = +0.68 0 = 1.05 +p y P y =-1.05 Use pythagorean equation to get p=1.25 kgm/s @57 degrees below +x axis. Use p=mv to get final vel of Mass 2: v=2.5 m/s @57 degrees below +x axis