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4.2 – Energy & Conservative Forces

What is Energy? • Energy is the ability or capacity to do work ○ Work is the process of moving an object • Energy is the ability or capacity to move an object Energy Transformations • Energy can be transformed from one type to another • You can transfer energy from one object to another by doing work • Work-Energy Theorem ○ Work done on a system by an external force changes the energy of the system Kinetic Energy • Kinetic Energy is energy of motion ○ The ability or capacity of a moving object to move another object • K=1/2 〖mv〗^2 Potential Energy • Potential Energy (U) is energy an object possesses due to its position or state of being ○ Gravitational Potential Energy (Ug) is the energy an object possesses because of its position in a gravitational field ○ Elastic Potential Energy (Us) ○ Chemical Potential Energy ○ Electric Potential Energy ○ Nuclear Potential Energy • A single object can have only kinetic energy, as potential energy requires an interaction between objects Internal Energy • The internal energy of a system include the kinetic energy of the objects that make up the system and the potential energy of the configuration of the objects that make up the system • Changes in a system's structure can result in changes in internal energy Gravitational Potential Energy (Ug) Conservative Forces • A force in which the work done on an object is independent of the path taken is known as conservative force • A force in which the work done moving along a closed path is zero • A force in which the work done is directly related to a negative change in potential energy (W=-ΔU) Conservative Forces Non-Conservative Forces Gravity Elastic Forces Friction Drag Air Resistance Coulombic Forces Newton's Law of Universal Gravitation • The gravitational force of attraction between any two objects with mass • F_g=−(〖Gm〗_1 m_2)/r^2 r ̂ Force from Potential Energy Summary

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4.3 – Conservation of Energy

Conservation of Mechanical Energy • Consider a single conservative force doing work on a closed system • ∵W_F=ΔK, W_F=−ΔU • ∴ΔK+ΔU=0 • ∴K_i+U_i=K_f+U_f Non-Conservative Forces • Non-conservative forces change the total mechanical energy of a system, but not the total energy of a system • Work done by a non-conservative force is typically converted to internal (thermal) energy • E_total=K+U+W_NC • E_mecℎ=K+U 2002 Free Response Question 3 2007 Free Response Question 3 2010 Free Response Question 1 2013 Free Response Question 1

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4.4 – Power

Defining Power • Power is the rate at which work is done • Power is the rate at which a force does work • Units of power are joules/second, or watts • P_avg=ΔW/Δt • P=dW/dt=(F ⃗⋅dr ⃗)/dt=F ⃗⋅(dr ⃗)/dt=F ⃗⋅v ⃗ 2003 Free Response Question 1

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5.1 – Momentum & Impulse

Momentum • Momentum is a vector describing how difficult it is to stop a moving object • Total momentum is the sum of individual momenta • p ⃗=mv ⃗ • Units are kg·m/s or N·s Example 1: Changing Momentum • An Aichi D3A bomber mass (3600 kg) departs from its aircraft carrier with a velocity of 85 m/s due east. What is its momentum? • After it drops its payload, its new mass is 3000 kg and it attains a cruising speed of 120 m/s . What is its new momentum? Impulse • As you observed in the previous problem, momentum can change • A change in momentum is known as an impulse (J) • J ⃗=Δp ⃗ Example 2: Impulse • The D3A bomber, which had a momentum of 3.6e5 kg·m/s, comes to halt on the ground. What impulse is applied? Relationship Between Force and Impulse • F ⃗=ma ⃗=m (dv ⃗)/dt=d/dt (mv ⃗ )=(dp ⃗)/dt Example 3: Force from Momentum • The momentum of an object as a function of time is given by p=kt2, where k is a constant. What is the equation for the force causing this motion? Impulse-Momentum Theorem Example 4: Impulse-Momentum • A 6-kg block, sliding to the east across a horizontal, frictionless surface with a momentum of 30 kg·m/s, strikes an obstacle. The obstacle exerts an impulse of 10 N·s to the west on the block. Find the speed of the block after the collision. Example 5: Water Gun • A girl with a water gun shoots a stream of water than ejects 0.2 kg of water per second horizontally at a speed of 10 m/s. What horizontal force must the girl apply on the gun in order to hold it in position? Impulse from F-t Graphs • Impulse is the area under a Force-time graph • Impulse is equivalent to a change in momentum Example 6: Impulse from Force • A force F(t)=t3 is applied to a 10 kg mass. What is the total impulse applied to the object between 1 and 3 seconds?

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5.2 – Conservation of Linear Momentum

Collisions and Explosions • In the case of a collision or explosion, if you add up the individual momentum vectors of all the objects before the event, you'll find that they are equal to the sum of momentum vectors of the objects after the event • Written mathematically, the law of conservation of linear momentum states • p ⃗_initial=p ⃗_final Solving Momentum Problems 1. Identify all the objects in the system 2. Determine the momenta of the objects before the event. Use variables for any unknowns 3. Determine the momenta of the objects after the event. Use variables for any unknowns 4. Add up all the momenta from before the event and set equal to the momenta after the event 5. Solve for any unknowns Types of Collisions • Elastic collision ○ Kinetic energy is conserved • Inelastic collision ○ Kinetic energy is not conserved Example 1: Traffic Collision • A 2000-kg car traveling 20 m/s collides with a 1000-kg car at rest. If the 2000-kg car has a velocity of 6.67 m/s after the collision, find the velocity of the 1000-kg car after the collision Example 2: Collision of Two Moving Objects • On a snow-covered road, a car with a mass of 1100 kg collides head-on with a van having a mass of 2500 kg traveling at 8 m/s • As a result of the collision, the vehicles lock together and immediately come to rest. • Calculate the speed of the car immediately before the collision Example 3: Recoil Velocity • A 4-kg rifle fires a 20-gram bullet with a velocity of 300 m/s. Find the recoil velocity of the rifle Example 4: Atomic Collision • A proton (mass=m) and a lithium nucleus (mass=7m) undergo an elastic collision as shown below. • Find the velocity of the lithium nucleus following the collision Example 5: Collisions in Multiple Dimensions • Bert strikes a cur ball of mass 0.17 kg , giving it a velocity of 3 m/s in the x-direction. When the cue ball strikes the eight ball (mass=0.16kg), previously at rest, the eight ball is deflected 45 degrees from the cur ball's previous path, and the cue ball is deflected 40 degrees in the opposite direction. Find the velocity of the cue ball and the eight ball after the collision 2001 Free Response Question 1 2002 Free Response Question 1 2014 Free Response Question 1

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