Momentum

Conservation of Momentum

⚛ Imagine a large cart moving along a frictionless track at a constant speed. This, of course, means that no net forces act on the cart. ⚛ Now run this cart into a second, smaller cart that is initially at rest on the track. ⚛ Which is the greater force: the force of the large cart on the small one or the force of the small cart on the large one? Careful...remember Newton's third law? The two carts apply the same force on each another. ⚛ They also apply equal but opposite reaction forces on each other for an equal amount of time, which means that both cars must experience the same impulse; they have the same horizontal net force over the same time. ⚛ Impulse equal to the change in momentum so both cars must also experience the same change to their momentum during the collision. ⚛ Since the forces are in opposite directions, the changes in the momentum will be in opposite directions; however much momentum one object loses during a collision, the other object must gain an equal amount of momentum. ⚛ In other words, as long as no external forces act on a system, the total momentum in the system must stay constant; the total momentum before a collision will equal the total momentum after the collision.

Lab Objectives

⚛ Lab: Identify the relationship between the momentum in a system before a collision and the momentum in a system after a collision ⚛ Lab: Discuss the effects of a collision on total system kinetic energy

Linear Momentum Equation

⚛ Momentum is directly proportional to the object's mass and also its velocity. ⚛ Thus the greater an object's mass or the greater its velocity, the greater its momentum. ⚛ Momentum 𝐩 is a vector having the same direction as the velocity 𝐯. ⚛ The SI unit for momentum is kg·m/s

perfectly inelastic collision

A collision in which the objects stick together is sometimes called "perfectly inelastic."

elastic collision

An elastic collision is one that conserves internal kinetic energy.

A 30 kg rocket is initially at rest on the ground. The rocket experiences an upward 400 N of force from the engine during launch. If the engine fires for 6.0 seconds, what is the final speed of the rocket? (Assume that the mass of the fuel is negligible.)

An important part of building and launching model rockets is to purchase the correct size engine for the rocket. Model rocket engines are rated by a letter system that indicates how much impulse a particular engine can apply to any given rocket. For example, an A-class engine will apply between 1.26 to 2.5 N-s of impulse whereas an H-class engine will apply between 160 to 320 N-s of impulse. The packaging on the rocket engine usually also indicates the engine burn time. Rocket hobbyists can then determine which impulse and burn time will result in the amount of thrust needed for a particular rocket. (According to Newton's second law, the engine thrust must be greater than the weight of the rocket if the rocket is to get off the ground.)

inelastic collision

An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).

Collisions

Caused by momentum and impulse

A 5.0 kg box moving with an initial velocity of 3.0 m/s slows to a speed of 1.0 m/s due to friction. What was the box's change in momentum?

Changes in momentum can occur via changes in either the mass or the velocity. In this course, however, only constant mass systems will be considered.

Three point masses start from rest at a position very nearly (0,0). When released from rest, they apply an initial repelling force on each other. Particle A has a mass of 0.1 kg and moves at a speed of 2 m/s to the west. Particle B has a mass of 0.2 kg and moves at a speed of 3 m/s at an angle 30° below the horizontal. Particle C has a mass of 0.3 kg. What is the velocity of particle C (speed and direction)?

Draw a diagram of the scenario. Guess the direction of the unknown velocity vector arrow.

Elastic and inelastic collisions Impacts and linear momentum

Elastic: Kinetic Energy is conserved Inelastic: Kinetic Energy is not conserved

internal kinetic energy

Internal kinetic energy is the sum of the kinetic energies of the objects in the system.

An astronaut on a space-walk has become untethered. The astronaut and her equipment come to a total mass of 150 kg. She is initially floating with zero velocity relative to her ship. In order to get to her ship, she throws some of her equipment in a direction away from the ship. The equipment has a mass of 5.0 kg and a velocity of 6 m/s. What is the resulting speed of the astronaut?

It can sometimes help to sketch out the scenario (boxes are an easy artistic choice). Add arrows to show the direction of any motion. Remember that arrows to the left mean a negative velocity. Add an equal sign between the before and after sketches and then add plus signs between all the other boxes. Write "mv" under each box. Congratulations, you have your equation!

A 0.5 kg toy car traveling to the right on a frictionless track with an initial velocity of 4 m/s hits a 0.25 kg toy car initially at rest on the track. The two do not stick together. The final speed of the larger car is 2.0 m/s to the right. What is the final speed of the second car?

It should be noted that momentum is ONLY conserved in a collision as long as there are no outside forces acting on the system. For example, if a ball is thrown to the right against a wall, the ball will bounce back to the left but the wall will not move to the right. The system (ball + wall) started with momentum to the right before the collision but ended with momentum to the left after the collision. The momentum of the ball + wall system is not conserved. This is because the wall has outside forces acting on it (other walls, the floor, the roof, etc).

Lab Momentum

Momentum is the product of an object's mass and velocity. If either the mass or the velocity changes, the momentum also changes. In this lab you will explore the relationship between the total system momentum before a collision and the total system momentum after a collision. You will also explore the effects that collisions have on the kinetic energy of a system.

Notice in the last example that the smaller car experienced a larger change to its velocity. Why is this? The momentum lost by the large car must equal the change in momentum of the smaller car:

Notice in the last example that the smaller car experienced a larger change to its velocity. Why is this? The momentum lost by the large car must equal the change in momentum of the smaller car:

The final velocity of combined Ball A and Ball B is 2.5 m/s at a direction of 18.4° above the x-axis.

That's it! This process is identical to the one used for one direction momentum conservation; the only real difference is that the process is completed twice (once in each direction) and the results combined using the same vector techniques used throughout the course. The only thing that makes conservation in two directions tricky is just keeping track of all the directions. Also, be very careful here; it is the momentum that is conserved in the collision—NOT the velocities.

A 0.145 kg baseball approaches a batter with an initial speed of 30 m/s. The batter hits the ball, and the ball leaves the bat moving in the opposite direction with a speed of 20 m/s. If the ball and the bat were in contact for 0.01 s, what was the force of the bat on the ball?

The problem does not specify the initial direction but the problem does note that the ball experienced a change in direction. If we assume the initial direction was positive and the final direction was negative then:

Momentum

The product of an object's mass and velocity

elastic collision & internal kinetic energy

We start with the elastic collision of two objects moving along the same line—a one-dimensional problem. An elastic collision is one that also conserves internal kinetic energy. Internal kinetic energy is the sum of the kinetic energies of the objects in the system. Figure 8.6 illustrates an elastic collision in which internal kinetic energy and momentum are conserved.

∑J = ∑Ft

Where J is the impulse, F is the force on the object, and t is the amount of time the force is experienced. The units of impulse are N-s.

impulse

change in momentum

Book Sections

⚛ 8.1 Linear Momentum and Force ⚛ 8.2 Impulse ⚛ 8.3 Conservation of Momentum ⚛ 8.4 Elastic Collisions in One Dimension ⚛ 8.5 Inelastic Collisions in One Dimension ⚛ 8.6 Collisions of Point Masses in Two Dimensions

Lesson Sections

⚛ 8.1 Momentum and Impulse ⚛ 8.2 Conservation in One Direction ⚛ 8.3 Conservation in Two Directions

Impulse

⚛ A conceptual framework can be built for this term by thinking through the process of how someone might catch a raw egg thrown at them. ⚛ The person catching the raw egg has two choices: stop the egg quickly or stop the egg slowly. In order to stop the egg quickly, the person would apply a large force for a short amount of time. This runs the risk of breaking the egg. The person could instead apply a small force to the egg for a longer amount of time by moving their hands back with the egg as they catch it. This method is less likely to break the egg. ⚛ The product of force and time in both examples is called impulse. From a conceptual standpoint, impulse can be thought of as a measurement of how much effort is put into making an object speed up or slow down. In the previous example, the largest impulse would be created by applying a large force to the egg over a long time period. Of course, this would not only break the egg but create bits of egg shrapnel.

Elastic vs Inelastic Collisions

⚛ All collisions can be classified as either elastic or inelastic collisions. ⚛ In an elastic collision, the total kinetic energy of the system is conserved. This means that the total kinetic energy of the system remains constant; the kinetic energy that the system has before a collision is equal to the kinetic energy the system has after the collision. ⚛ In an inelastic collision, the total kinetic energy of the system is NOT conserved. Some kinetic energy is lost during the collision. Usually the lost kinetic energy is the result of internal friction, heat, deformation of the objects, etc. ⚛ It should be noted that a true elastic collision is an ideal; they don't actually exist. However, many examples of collisions are so close to the ideal that scientists treat them as such. The collision of gas particles is one example. (You may have studied ideal gas laws in your chemistry classes.) ⚛ Remember that, as long as no outside forces act on the system, the total momentum will be conserved regardless of whether the collision is elastic or inelastic.

impulse summary

⚛ Another name for change in momentum is impulse. ⚛ Impulse is the product of net force and the time for which the net force is applied. ⚛ Since impulse and change in momentum are the same thing, this also means the following: m∆v = ∑Ft ⚛ In other words, if an object experiences a net force, it will experience a change in momentum.

quantitative perspective

⚛ Ball A has a mass of 2.0 kg and moves to the east with a speed of 3 m/s. Ball B has a mass of 0.5 kg and moves to the north with a speed of 4 m/s. When the two collide, they stick together. What is the combined velocity after the collision (speed and direction)? ⚛ First, sketch and label the scenario. These problems can be very difficult to solve without a picture reference.

Lesson Objectives

⚛ Define and calculate momentum ⚛ Use the impulse-momentum theorem to calculate forces and changes in velocity during a collision ⚛ Use conservation of momentum to predict the velocities of one or more objects before or after a collision ⚛ Compare and contrast elastic and inelastic collisions

Collisions

⚛ If objects in a system collide, the total momentum of the system will remain constant as long as no net forces act on the system. This concept is called the law of conservation of momentum. Momentum is conserved in all directions during collisions between objects. ⚛ Collisions where the kinetic energy of a system is conserved are called elastic collisions. If the kinetic energy is not conserved, the collision is inelastic.

Conservation in Two Directions

⚛ Imagine a ball colliding with an identical ball that is initially at rest. As long as friction is negligible and the collision is elastic, the first ball will transfer all of its momentum to the second ball. The first ball will stop completely, while the second ball continues with the same starting velocity the first ball had. ⚛ Now, what if the first ball does not hit the second ball straight-on but with a glancing blow? One ball will travel down and forward while the other will travel up and forward:

momentum!

⚛ Momentum is defined by an equation, so it can be difficult to find a good, conceptual definition for it. One way to think about momentum is as a measurement of how difficult it is to stop a particular moving object. ⚛ If an object has a large inertia (measured by mass), it will be difficult to change the motion of the object. Thus, inertia and momentum are related. ⚛ If an object has a large velocity, it will also be difficult to stop. The larger the velocity, the larger the net force must be in order to stop the object in a given time period. This means a relationship between velocity and momentum exists as well.

Momentum Summary

⚛ Momentum is the product of mass and velocity. ⚛ A change in momentum can be obtained by changing either the mass or the velocity but this only considers changes in momentum due to changes in velocity.

Why is this?

⚛ Since the collision is a bit offset, the force will also be a bit offset; the collision force is not a horizontal force. The ball initially at rest will get pushed down and to the right. ⚛ This idea explains why the ball initially at rest moves down and to the right, but why does the first ball move up and to the right? ⚛ Remember that momentum must be conserved. This is true for momentum in all directions. ⚛ The total horizontal momentum of the first ball will equal the combined horizontal momentum values of the two balls after the collision. The total vertical momentum of the first ball will equal the combined vertical momentum values of the two balls after the collision. Since no vertical momentum was present before the collision, the vertical momentum components of the two balls after the collision must cancel. Since the ball initially at rest gained a negative vertical component, the first ball had to gain an equal positive vertical component.

Linear Momentum

⚛ The scientific definition of linear momentum is consistent with most people's intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. ⚛ Linear momentum is defined as the product of a system's mass multiplied by its velocity. ⚛ In symbols, linear momentum is expressed as p = mv

The impulse-momentum theorem

⚛ These two equations should look familiar. The left side of the equation is the net impulse that acts on an object, and the right side is the object's change in momentum. ⚛ This equation shows that net impulse is equal to an object's change in momentum; it is called the impulse-momentum theorem. The textbook does not use any symbols to represent impulse because it always treats net impulse as a change in momentum, Δp. There is nothing wrong with this since the two values are always equal. ⚛ It's important to remember that the impulse-momentum theorem is just a rearrangement of Newton's second law. This concept isn't anything new; it's just a different way of thinking about the same ideas. ⚛ The impulse-momentum theorem is particularly useful when examining collisions between objects.

momemtum equation

⚛ Where p is momentum, m is mass (inertia), and v is velocity. ⚛ The units of momentum are the same as the units of mass x the units of velocity or kg-m/s. ⚛ A vector measurement, which means it has both magnitude and direction. ⚛ Unsurprisingly, the direction of the momentum vector is the same direction as the velocity vector. ⚛ In other words, an object's momentum is in the same direction as its motion.