Impulse and Momentum AP Physics C

Momentum The product of a particle’s mass and velocity is called the momentum of the particle Momentum is a vector, with units of kg m/s. A particle’s momentum vector can be decomposed into x- and y-components. We will use momentum to help analyze what happens during a collision.

Collisions A collision is a short-duration interaction between two objects. During a collision, it takes time to compress the object, and it takes time for the object to re-expand. The duration of a collision depends on the materials. When kicking a soccer ball, the amount by which the ball is compressed is a measure of the magnitude of the force the foot exerts on the ball. The force is applied only while the ball is in contact with the foot. The impulse force is a large force exerted during a short interval of time.

Impulse-Momentum Theorem

Impulse-Momentum Theorem Intuitively we know that giving a kick to a heavy object will change its velocity much less than giving the same kick to a light object. Using Newton’s second law, we can derive what is called the impulse-momentum theorem, which relates force to change in velocity.

Example A 2.0 kg object moving to the right with speed 0.50 m/s experiences the force shown. What are the object’s speed and direction after the force ends?

Example 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?

Hedgehogs and DeAndre Hopkins Understand Physics, therefore they are awesome The average force needed to stop an object is inversely proportional to the duration of the collision. If the duration of the collision can be increased, the force of the impact will be decreased. This is used to make cars safer during collisions, and it is even a tactic used by hedgehogs and NFL receivers. Hedgehogs can puff their spines to cushion their fall, which causes their momentum to change over a longer period of time, therefor decreasing the force during impact. Receivers bring the ball in while catching it, which also decreases the force of impact, making it easier to catch.

Example A light plastic cart and a heavy steel cart are both pushed with the same force for 1.0 s, starting from rest. After the force is removed, which cart has the greater momentum, if any?

Conservation of Momentum

Total Momentum of a System If there is a system of particles moving, then the system as a whole has an overall momentum. The total momentum of a system of particles is the vector sum of the momenta of the individual particles:

Conservation of Momentum If we define a system where there are no external forces acting on the system, then the total momentum in the system does not change. This means that momentum of the system is conserved. Each object in the system exerts internal forces on each other, but these forces cancel each other out because of Newton’s third law.

Individual Momentum in a Collision What we must keep in mind is that the momentum of each object in our system is not conserved during a collision. When two balls collide, they will often change directions after the collision. This tells us that each ball experienced a change in momentum, but the system as a whole did not. The total momentum of the system (the vector sum of all momentums in the system) is conserved.

Example You awake in the night to find that your living room is on fire. Your one chance to save yourself is to throw something that will hit the back of your bedroom door and close it, giving you a few seconds to escape out the window. You happen to have both a sticky ball of clay and a super-bouncy Superball next to your bed, both the same size and same mass. You’ve only time to throw one. Which will it be? Your life depends on making the right choice!

Collisions

Types of Collisions There are 4 types of collisions that we will study in this unit: Elastic Inelastic Perfectly Inelastic Explosion In all of these cases, we will say that momentum of the system is conserved. The differences will be whether or not kinetic energy is conserved, and how many objects we start and end with.

Elastic Collisions An elastic collision is one in which both momentum and kinetic energy of the system is conserved. In an elastic collision we have multiple objects before the collision and multiple objects after the collision. Although no collision is truly elastic, collisions between very hard objects such as pool balls or ball bearings come very close to being elastic. 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝑝 𝑎𝑓𝑡𝑒𝑟 𝐾𝐸 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝐾𝐸 𝑎𝑓𝑡𝑒𝑟

Inelastic Collisions An inelastic collision is one in which momentum of a system is conserved, but the kinetic energy of the system is not conserved. In an inelastic collision we have multiple objects before the collision and multiple objects after the collision. The kinetic energy lost in an inelastic collision is converted into thermal energy. Most collisions are inelastic. 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝑝 𝑎𝑓𝑡𝑒𝑟 𝐾𝐸 𝑏𝑒𝑓𝑜𝑟𝑒 > 𝐾𝐸 𝑎𝑓𝑡𝑒𝑟

Perfectly Inelastic Collisions A perfectly inelastic collision is one in which momentum of a system is conserved, but the kinetic energy of the system is not conserved. In an inelastic collision we have multiple objects before the collision and one object after the collision. This means that the objects stick together. The kinetic energy lost in an inelastic collision is converted into thermal energy. 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝑝 𝑎𝑓𝑡𝑒𝑟 𝐾𝐸 𝑏𝑒𝑓𝑜𝑟𝑒 > 𝐾𝐸 𝑎𝑓𝑡𝑒𝑟

Explosions An explosion is one in which momentum of a system is conserved, but the kinetic energy of the system is not conserved. In an explosion we have one object before the collision and multiple objects after the collision. This means that one object splits into two (or possibly more). The kinetic energy produced in an explosion comes from the potential energy that was originally in the system. 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝑝 𝑎𝑓𝑡𝑒𝑟 𝐾𝐸 𝑏𝑒𝑓𝑜𝑟𝑒 < 𝐾𝐸 𝑎𝑓𝑡𝑒𝑟

In summary Collision Type Elastic Inelastic Perfectly Inelastic Explosion Momentum Conserved Yes Kinetic Energy Conserved No Objects before the collision 2 1 Objects after the collision

How to Solve Collision Problems Momentum of the system is conserved in collision problems. You often are asked to find the velocity of an object before or after a collision, use conservation of momentum to find that. Remember that momentum and velocity are vectors, watch your signs! Explosion Perfectly Inelastic

Example The two boxes are on a frictionless surface. They had been sitting together at rest, but an explosion between them has just pushed them apart. How fast is the 2-kg box going?

Example The 1-kg box is sliding along a frictionless surface. It collides with and sticks to the 2-kg box. Afterward, the speed of the two boxes is:

Example 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?

Example The two boxes are sliding along a frictionless surface. They collide and stick together. Afterward, the velocity of the two boxes is:

Two Dimensional Collisions When the motion of the collisions occur in two dimensions, we must solve for each component of the momentum: In these collisions, the individual momenta can change but the total momentum does not.

Example Two pucks of equal mass 100 g collide on an air hockey table. Neglect friction. Prior to the collision, puck 1 travels in a direction that can be considered the +x-axis at 1 m/s, and puck 2 travels in the –y-direction at 2 m/s prior to the collision. After the collision, puck 2 travels 30 degrees above the +x-direction (between +x and +y) at 0.8 m/s. What is the velocity (direction and speed) of puck 1 after the collision?