The Immaculate Reception

AP Physics C: Mechanics MOMENTUM AP Physics C: Mechanics

What is Momentum? What is its definition? How do we calculate it? When do we use this term? Why was this word invented? What do we already know about it? What do we want to know about it?

What is Momentum? Momentum: “quantity of motion” -Newton What is its definition? Momentum: “quantity of motion” -Newton Momentum: “mass in motion” Momentum: the product of an object’s mass and its velocity Momentum: It is a vector! Momentum: is sometimes called linear momentum

What is Momentum? How do we calculate it? What are its units? If object is moving in arbitrary direction: What are its units?

What do we know about momentum?

What is Momentum? Why was this word invented? When do we use this term? We are yet to make a distinction between a rhino moving at 5m/s and a hummingbird moving at 5m/s. Thus far, how have we handled forces that are only briefly applied such as collisions? (we pretended that doesn’t happen) Some believed that this quantity is conserved in our universe.

How is momentum related to other physics concepts that we have already studied? The time rate of change of linear momentum of a particle is equal to the net force acting on the particle. We will soon see that it has many things in common with Energy, Newton’s 3rd law, and The Calculus.

Pause to think about calculus concepts: Why is a derivative involved? What does this say about the slope of a momentum-time graph? The area under which graph might be meaningful? So, how might an integral be involved? Momentum may be changing non-uniformly with time The slope of a momentum-time graph is net force! The area under a force-time graph is a change in momentum! The integral of force with respect to time is a change in momentum!

Pause to think about calculus concepts: The integral of force with respect to time is a change in momentum! We call the left-hand side of this equation the IMPULSE of the force

Pause to think about calculus concepts: The slope of a momentum-time graph is net force! The area under a force-time graph is a change in momentum or an impulse

Impulse-Momentum Theorem: The impulse of a force F equals the change in momentum of the particle. This is another way of saying that a net force must be applied to change an objects state of motion. Why does this look different from the last equation? Because the force might be constant!

A few things about IMPULSE: It is a vector in the same direction as the change in momentum. It is not a property of an object! It is a measure of the degree to which a force changes a particles momentum. We say an impulse is given to a particle. What are its units? From the equation we see that they must be the same as momentum’s units (kgm/s). Impulse approximation: assume the force is applied only for an instant and that it is much greater than other forces present.

Another question please…

To stop a speeding train: Explain these videos in physics terms.

Quick Conceptual Quiz Can a hummingbird have more momentum than a rhino? Why might an out of control truck hit a haystack or barrels and pile of sand as opposed to a wall as an emergency stop? How is a ninja’s ability to break stacks of wood related to impulse and momentum? What good is it to know an object’s momentum?

Question 2: If a boxer is able to make his impact time 5x longer by “riding” with the punch, how much will the impact force be reduced? By 5x

When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard floor? No – the same impulse – the force exerted on the dish is less because the time of momentum change increases.

Bend knees when jumping Gymnasts and wrestlers use mats Examples Examples of Increasing Impact Time to decrease Impact Force: Bend knees when jumping Gymnasts and wrestlers use mats Glass dish falling on carpet rather than concrete Acrobat safety net Other examples???

Observing changes in momentum:

Consider two particles that can interact, but are otherwise isolated form their surroundings. What do we know about a collision between these two particles? Newton’s law says that they exert equal and opposite forces on each other regardless of comparative size (mass). Is it possible for one particle to be in contact with the second particle for a longer period of time than the second on the first? No, so the impulse imparted on each must be the same. THEREFORE…

The particles must undergo the same changes in momentum! Let’s look at this mathematically.

It means that the total momentum of the system is constant over time. What does it mean, conceptually, for a time derivative of momentum to be zero It means that the total momentum of the system is constant over time. aka Momentum is Conserved!

The Law of Conservation of Momentum When two isolated, uncharged particles interact with each other, their total momentum remains constant. OR The total momentum of an isolated system at all times equals its initial momentum (before and after collisions).

Find the rebound speed of a 0 Find the rebound speed of a 0.5 kg ball falling straight down that hits the floor moving at 5m/s, if the average normal force exerted by the floor on the ball was 205N for 0.02s.

A mass m is moving east with speed v on a smooth horizontal surface explodes into two pieces. After the explosion, one piece of mass 3m/4 continues in the same direction with speed 4v/3. Find the magnitude and direction for the velocity of the other piece. A) v/3 to the left B) The piece is at rest. C) v/4 to the left D) 3v/4 to the left E) v/4 to the right

How good are bumpers? A car of mass 1500kg is crash-tested into a wall. It hits the wall with a velocity of -15m/s and bounces off with a velocity of 2.6m/s. If the collision lasts for 0.15s, what is the average force exerted on the car?

Types of Collisions Energy is always conserved but may change types (mv2/2, mgh, kx2/2 etc). There is only one type of momentum (mv). We identify collisions based upon their conservation of kinetic energy. Inelastic kinetic energy is NOT constant Elastic kinetic energy IS constant

Inelastic Collisions These collisions are considered PERFECT when the objects collide and combine to move as one object. Inelastic Objects bounce but may be deformed so kinetic energy is transformed. Perfectly Inelastic Objects stick together

Perfectly Inelastic Collisions:

Elastic Collisions (ideally)

For elastic collisions, find an expression for relative speed of the objects before and after collision. From momentum conservation…

For elastic collisions, find an expression for final speed in terms of initial speeds and mass. From kinetic energy conservation… Divide out ½ and move like mass terms to the same side so mass can be factored out… Factor difference of squares…

Combine our two results… The relative speed of the two objects before an elastic collision equals the negative of their relative speed after.

Solve for final speeds in terms of initial speeds and mass.

Two-dimensional Collisions Set coordinate system up with x-direction the same as one of the initial velocities Label vectors in a sketch Write expressions for components of momentum before and after collision for each object v1f v1fsinθ v1fcosθ v1i θ φ v2fcosφ -v2fsinφ v2f

The types of collisions are treated the same mathematically.