Impulse and Momentum AP Physics 1

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. 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚=𝑝=𝑚𝑣

Impulse When objects collide, they exert forces on each other. This changes the momentum of each object. Change in momentum = Impulse. This is known as the impulse-momentum theorem. Impulse is also a vector, and it points in the direction of net force. The units for Impulse are Ns. The variable we use for impulse is J… because science. 𝐹 𝑛𝑒𝑡 =𝑚𝑎 𝑎= ∆𝑣 ∆𝑡 𝐹 𝑛𝑒𝑡 =𝑚 ∆𝑣 ∆𝑡 𝐹 𝑛𝑒𝑡 ∆𝑡=𝑚∆𝑣→ 𝐹 𝑛𝑒𝑡 ∆𝑡=∆𝑝=𝐽

Graphical Analysis If we graph force vs. time for a collision, we find that the area under the curve represents the impulse delivered to the object. Remember that because impulse = change in momentum, the area under the curve for this graph also represents the change in momentum for the object.

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?

Cars and Hedgehogs and Physics Cars are made of “crunchable” materials, that way when they get in a wreck the momentum is changed over a long period of time. This decreases the force on the driver! Hedgehogs puff out their needles when falling to cushion their fall. The amount of time it takes to change their momentum is larger, therefor decreasing the force on the hedgehog. Plus they are adorable. 𝐹∆𝑡=∆𝑝 𝐹= ∆𝑝 ∆𝑡 Force and time are inversely proportional!

Collisions Consider two carts moving towards each other on a frictionless track. When the carts collide, the exert equal and opposite forces on each other (Newton’s Third Law). Because the carts are in contact for equal amounts of time, the impulse delivered to each cart is the same. 𝐹 1 =− 𝐹 2 𝑡 1 = 𝑡 2 𝐹 1 𝑡 1 =− 𝐹 2 𝑡 2 → 𝐽 1 =− 𝐽 2

Collisions If the impulses are equal to each other, the change in momentum for each cart is the same. 𝐽 1 =− 𝐽 2 ∆ 𝑝 1 =−∆ 𝑝 2 𝑚 1 ∆𝑣= 𝑚 2 ∆𝑣 𝑚 1 𝑣 1 − 𝑣 𝑜1 = 𝑚 2 𝑣 2 − 𝑣 𝑜2 𝑚 1 𝑣 𝑜1 + 𝑚 2 𝑣 𝑜2 = 𝑚 1 𝑣 1 + 𝑚 2 𝑣 2 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝑝 𝑎𝑓𝑡𝑒𝑟 This means that the momentum of the system is conserved!

Conservation of Momentum In the absence of a net external force, the momentum of a system is conserved. This means that if we add the momenta of each object before the collision, it should equal the sum of the momenta after the collision. Watch your signs! Momentum is a vector, make sure you use the correct signs when doing calculations.

Momentum and Systems The action/reaction pairs between each object are internal to our system, so they do not change the momentum of the system. If there is an external force on the system, it will change the momentum of the system. This means there is an impulse delivered to the system. We generally choose our system so that the momentum of the system is conserved.

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

Example 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?

Momentum in Two Dimensions Momentum is a vector, so it can be broken into components. In head on collisions, this is not necessary, but what if the collision is not head on? Momentum is conserved in both the x and y direction, and these momenta are independent of one another. Fun fact: Understanding 2 dimensional collisions will make you a lot better at pool. 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒,𝑥 = 𝑝 𝑎𝑓𝑡𝑒𝑟,𝑥 𝑝 𝑏𝑒𝑓𝑜𝑟𝑒,𝑦 = 𝑝 𝑎𝑓𝑡𝑒𝑟,𝑦