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Momentum

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Impulse

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A collision is a short-duration interaction between two objects. Collisions may appear instantaneous, but their duration, however small, is significant.

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Impulse Applied force increases and then decreases in magnitude throughout the collisions duration. (Think kicking a soccer ball). A large force like this exerted during a short interval of time is called an impulsive force.

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Impulse An impulsive force of greater magnitude (taller force curve) or a force applied over a longer duration (wider force curve) has a greater effect. A taller or wider force curve also has a larger area under the curve. This is area is called the impulse, J.

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Impulse

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Momentum and the Impulse- Momentum Theorem

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Momentum Effect of an impulsive force also depends on the mass of the object. e.g., kicking a heavy object will change its velocity much less than giving the same kick to a light object. Consider an object moving with initial velocity, v o. You kick this object and deliver an impulse J = F avg t After impulse, the object now moves with a final velocity, v. How is this final velocity related to the initial velocity?

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Momentum

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The Impulse-Momentum Theorem

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EXAMPLE 1 Calculating the change in momentum 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 balls momentum? What is the impulse delivered to it by the wall?

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EXAMPLE 1 Calculating the change in momentum

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EXAMPLE 2 A well-hit ball A baseball of mass 0.14 kg has an initial velocity of 38 m/s as it approaches a bat. The bat applies a force that is much larger that the weight of the ball, and the ball departs from the bat with a final velocity of 58 m/s. (a) Determine the impulse applied to the ball by the bat. (b) Assuming that the time of contact is 1.6 x 10 -3 s, find the average net force exerted on the ball by the bat.

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EXAMPLE 2 A well-hit ball

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EXAMPLE 3 A Rain Storm During a storm, rain comes straight down with a velocity of - 15 m/s and hits the roof of a car perpendicularly. The mass of rain per second that strikes the car roof is 0.060 kg/s. Assuming that the rain comes to rest upon striking the car, find the average force exerted by the rain on the roof.

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EXAMPLE 3 A Rain Storm

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Stopping Objects

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If the duration of the collision can be increased, the force of the impact will be decreased. This is the principal used in most impact-lessening techniques, like the water barrels on the freeway, or bending your knees when coming down from free-fall.

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Total Momentum

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Conservation of Momentum

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Newtons Laws and Momentum Impulse-momentum theorem was derived from Newtons second law. It serves as an alternative way of looking at second law, but in the context of only one particle at a time. Consider now two objects instead. When two objects collide, they exert forces on each other in often complicated ways, so using only NSL to predict the behavior of these collisions would be difficult. Newtons third law provides a simpler way of predicting the outcome of a collision.

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Action-Reaction and Momentum Picture two balls headed toward each other, colliding and then bouncing apart. The forces during the collision, when the balls are interacting, are the action-reaction pair F 1 on 2 and F 2 on 1.

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Action-Reaction and Momentum During the collision, the impulse J 2 delivered to ball 2 by ball 1 is the average value of F 1 on 2 multiplied by the collision time t. Likewise, the impulse J 1 delivered to ball 1 by ball 2 is the average value of F 2 on 1 multiplied by t. F 1 on 2 and F 2 on 1 are an action-reaction pair equal in magnitude but opposite in direction, so J 1 = -J 2.

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Action-Reaction and Momentum Impulse-momentum theorem says p is equal to J, so p for either ball is also equal in magnitude but opposite in sign. In other words, if ball 1s momentum increases by a certain amount during the collision, ball 2s momentum will decrease by exactly the same amount. This implies that total momentum P = p 1 + p 2 of the two balls is unchanged by the collision. Because it doesnt change during the collision, we say momentum is conserved.

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Law of Conservation of Momentum The same conservation of momentum holds true for systems containing any number of objects. Forces that act only between particles within the system are called internal forces.

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Law of Conservation of Momentum

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EXAMPLE 4 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 Davids speed?

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EXAMPLE 4 Speed of ice skaters pushing off

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Explosions An explosion, where the particles of the system move apart after a brief, intense interaction, is the opposite of a collision. Explosive forces (expanding spring, expanding hot gases) are internal forces, so if the system is isolated, its total momentum during the explosion will be conserved.

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EXAMPLE 6 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?

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EXAMPLE 6 Recoil speed of a rifle

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Inelastic Collisions

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Sometimes colliding objects bounce off each other. This type of collision is known as an elastic collision. Colliding objects may also stick to each other. This is known as an inelastic collision.

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EXAMPLE 7 Speeds in an inelastic glider collision In a laboratory experiment, a 200g air-track glider and a 400 g air-track glider are pushed toward each other from opposite ends of the track. The gliders have Velcro tabs on their fronts so that they will stick together when they collide. The 200 g glider is pushed with an initial speed of 3.0 m/s. The collision causes it to reverse direction at 0.50 m/s. What was the initial speed of the 400 g glider?

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EXAMPLE 7 Speeds in an inelastic glider collision

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Momentum and Collisions in Two Dimensions

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EXAMPLE 8 Analyzing a peregrine falcon strike Peregrine falcons often grab their prey from above while both falcon and prey are in flight. A falcon, flying at 18 m/s, swoops down at a 45 angle from behind a pigeon flying horizontally at 9.0 m/s. The falcon has a mass of 0.80 kg and the pigeon a mass of 0.36 kg. What are the speed and direction of the falcon (now holding the pigeon) immediately after impact?

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EXAMPLE 8 Analyzing a peregrine falcon strike

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After the collision, the two birds move with a common velocity that is directed at an angle. X-component of the final momentum is