Sometimes (like on AP test) you will see the equation like this:

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1 Work, Energy & Momentum Notes Chapter 5 & 6 The two types of energy we will be working with in this unit are: (K in book KE): Energy associated with of an object. (U in book PE): Energy associated with the of an object. Vector Review: Adding Vectors: To add vectors at an angle, their x & y components are added and the Pythagorean theorem is used to determine the new resultant vector. Multiplying Vectors: Dot Product: A. B = Components of vectors in the same direction are multiplied. Answer is a scalar. Example: (draw the picture) With the dot product, it is important to write the dot Work Work: the dot product of force and displacement. In other words, work is a force times a displacement, but the force needs to be in the same direction as the displacement. x = displacement of object (your book uses d) θ = angle between the displacement & the force Units of work: Joule (J) 1 J = 1 N. m Sometimes (like on AP test) you will see the equation like this: The force must be in the as the movement. If not, you have to use a vector component of the force that is in the same direction. Hence the cosine. In order to do work, you must exert a force over a distance. The con may expend energy when he pushes on the wall, but if it doesn t move, no work is performed on the wall. Example: A man cleaning his apartment pulls a vacuum cleaner with a force of at an angle of A frictional force of magnitude retards the motion, and the vacuum is pulled a distance of 3.0 m. Calculate: a) the work done by the 50. N pull. b) the work done by the frictional force. c) the net work done on the vacuum by all forces acting on it. Work done by a varying force: Imagine pushing a heavy moving box through the carpeted living room and into the tiled kitchen. As the box moves from the carpet and onto the tile, it takes less force to keep the box moving. A graph of this scenario would look like: The under the Force vs Distance graph is the

2 Relating Work and Kinetic Energy (aka. The Work-Energy Theorem) W net = KE = Roughly speaking,. The net work done on an object by the force or forces acting on it is equal to the change in the kinetic energy of the object. o The speed of an object will if the net work done on it is. o The speed of an object will if the net work done on it is. Example: A car has a net forward force of 4500 N applied to it. The car starts from rest and travels down a horizontal highway. What are its kinetic energy and speed after it has traveled 100.m? (ignore friction and air resistance) Potential Energy Gravitational potential energy: the energy an object has due to its position in space. U = Gravitation Potential Energy (PE in your book) (J) m = mass of object (kg) g = 9.8 m/s 2 (on Earth) h = height of object (m) sometimes seen as y In working problems, choose the zero level for U (the pt. at which the grav. U=0) so that you can easily calculate the difference in U Example: A skier is at the top of a slope. At the initial point A, the skier is vertically above point B. Find the gravitational PE of the skier at A and B, and the difference in PE between the 2 pts. Work causes a change in Gravitational Potential Energy (U) Work done by the force of gravity causes a change in gravitational energy However, when the force of gravity is dong work on an object, the object is. A falling object its potential energy. Therefore, work done by the gravitational force is equal to the negative change in potential energy.

3 Law of Conservation of Energy: E i = E f Although energy cannot be destroyed, it can go places where we can never recover Although energy may be changed into a different form, the final value will be the same as the initial value (energy is not lost). Example: Calculate the KE and PE energy at each of the 5 spots on the ski run. Assume the skier starts from rest. (m = 51 kg) (First draw the route) Example: A diver weighing (mass = 77.0 kg) drops from a board 10 m above the surface of the pool of water. (a) Use the conservation of mechanical energy to find her speed at a point 5.00 m above the water surface. (b) Find the speed of the diver just before she strikes the water. Example: A sled and its rider together weigh. N. They move down a frictionless hill through a vertical distance of 10.0 m. Use conservation of energy to find the speed of the sled-rider system at the bottom of the hill, assuming the rider pushes off with an initial speed of. Conservative / Nonconservative Forces A force is if the work it does on an object between 2 points is of the path the object takes between the 2 points. (The work done depends only on the initial and final positions.) A force is if the work it does on an object moving between 2 points the path taken. (i.e. sliding friction)

4 Example: A 15 kg kid, initially at rest, slides down a high slide. Ideally, what his is final velocity at the bottom of the slide? If the final velocity is only 10.0 m/s. where did the extra energy go? Friction is an nonconservative force which used up some of the initial potential energy. To account for friction, we need to add back in the work done by (or energy lost by) friction. Note: Usually work done by friction is negative (because it is in the opposite direction). But since we are ADDING BACK IN the work done by friction it is positive. Example: How much work was done by friction as the kid (in the previous example) slid down the slide? Potential Energy Stored in a Spring: k = spring constant (N/m) x = Distance spring is compressed or stretched (m) Now we can add the potential energy of a spring into our conservation of energy equation: Example: A block of mass kg rests on a horizontal, frictionless surface. The block is pressed lightly against a spring, having a spring constant k=80.0 N/m. The spring is compressed a distance of 2.00 cm and released. Find the speed of the block at the instant it loses contact with the spring at the x = 0 position. Example: The launching mechanism of a toy gun consists of a spring of unknown spring constant. By compressing the spring a distance of m, the gun is able to launch a 20.0-g projectile to a maximum height of 20.0 m when fired vertically from rest. Determine the value of the spring constant. Example: A 5.0 kg block is given an initial velocity of 3.0 m/s at the top of a hill. Calculate the distance the spring will be compressed when it stops the block.

5 Simple Machines: Machines make work easier. Work In = Work Out + heat In a real machine, the efficiency must always be less than 100% (2 nd law of thermodynamics) Force x Distance = Force x Distance Machines make work easier. But, the amount of work stays the same. Power: The rate at which work is done. SI Units: English Units: Example: A car starts from rest and accelerates to a final velocity of m/s in a time of 15.0 s. Assume that the force of air resistance remains constant at a value of 500. N during this time. (a) Find the average power developed by the engine (express in watts and hp). (b) Find the instantaneous power when the car reaches its final velocity (in watts and hp). Example: An elevator has a mass of kg and carries a maximum load of 800. kg. A constant frictional force of N retards its motion upward. What minimum power must the motor deliver to lift the fully-loaded elevator at a constant speed of 3.00 m/s? (In watts and horsepower)

6 Momentum and Collisions (Chapter 6) Momentum and Impulse The momentum of an object is the product of its mass and velocity: If the resultant force F is zero, the momentum of the object does not change. Impulse-Momentum Theory: units of momentum: kg m/s If we exert a force on an object for a time interval Δt, the effect of this force is to change the momentum of the object from some initial value mv i to some final value mv f Example: A golf ball of mass 50. g is struck with a club. Assume that the ball leaves the club face with a velocity of +44m/s. (A) Estimate the impulse due to the collision. (B) Estimate the length of time of the collision and the average force on the ball. Applications of the impulse-momentum theory: Follow through in golf swing, batting, tennis. Catching a water balloon. Moving with the punch in boxing. Padding boxing gloves, goalie gloves in hockey, baseball mitts, inside of helmets Example: In a crash test, a car of mass 1500 kg collides with a wall. The initial of the car is 15.0 m/s east and the final velocity of the car is 2.6m/s west. If the collision lasts for s, find (A) Why does the car start off going east and end up going west? (B) the impulse due to the collision (C) the average force exerted on the car. Graphing Force vs Time: During most collisions the force isn t constant. If the force is graphed as a function of time, it would look like this: The area under the curve of the force vs time graph is the impulse (J) or change in momentum (Δp) This graph can be approximated as a triangle. What is the impulse during this collision?

7 Conservation of Momentum Law of conservation of momentum: when no external forces act on a system consisting of 2 objects, the total momentum of the system before the collision = the total momentum of the system after the collision. Example: A baseball player uses a pitching machine to help him improve his batting average. He places the 50.-kg machine on a frozen pond. The machine fires a 0.15-kg baseball with a speed of 36 m/s in the horizontal direction. What is the recoil velocity of the machine? Elastic and Inelastic Collisions Collisions For any type of collision, the momentum is. However, the total is generally conserved. Elastic Collisions Elastic collisions: both momentum and kinetic energy are conserved. Before: After: Inelastic Collisions Inelastic collisions: momentum (only) is conserved. Kinetic energy is not conserved. When 2 objects collide and stick together, the collision is PERFECTLY INELASTIC; in this case, their final velocities are the same. For perfectly inelastic collisions, Before: After: NOTE: THE ELASTIC AND INELASTIC EQUATIONS WILL NOT BE GIVEN ON ANY TEST. YOU NEED TO KNOW HOW TO FIGURE THEM OUT ON YOUR OWN!!!!

8 Example (perfectly inelastic): A large luxury car with a mass of kg stopped at a traffic light is struck from the rear by a compact car with a mass of 900. kg. The two cars become entangled as a result of the collision. If the compact car was moving at 20.0 m/s before the collision, what is the velocity of the entangled mass after the collision? Example (perfectly elastic): Two kids are playing marbles. The shooter rolls toward a stationary marble at 5.0 cm/s. They two collide and the smaller marble speeds away at 8.0 cm/s. What is the final velocity of the shooter? Assume the shooter is three times as massive as the smaller marble. Glancing Collisions In glancing collisions, the colliding masses rebound at some angle relative to the line of motion of the incident mass. Momentum is conserved along the direction and along the direction Example (perfectly inelastic): A kg car traveling east with a speed of 25.0 m/s collides at an intersection with a kg van traveling north at a speed of 20.0 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together.

9 Mixed Energy & Momentum Problems Most problems you will encounter will need a mixed of both energy & momentum equations. How to tell the difference between Energy & Momentum: Rule of Thumb: If the object stays the same it is a conservation of energy problem. If the object changes, it is a conservation of momentum problem (inelastic collision) If two objects bounce, it is a conservation of momentum problem (elastic collision) Springs can look like momentum problems because you start with one object (block and spring together) and end up with two (block and spring separate). However, if there is a spring, it is an energy problem. Example: A block with a mass of 25 kg starts from rest 6.0 m above the ground and slides down a frictionless ramp. At the bottom of the ramp, the block collides inelasticly with 10. kg toy car at rest. What is the final velocity of the block & car combo? Example: A 0.25 kg Nerf arrow is fired from spring-loaded gun at a 10.0 kg target sitting on a frozen (frictionless) pond. After impact, the arrow and target slide across the pond at 3.7 m/s. What is the spring constant of the Nerf gun if the spring was compressed 30.0 cm when the arrow was fired? (assume the arrow does not fall during flight) (Yes, I know this is a little unrealistic)