Physics 20 - Lesson 31: Conservation of Energy

Conservation of Energy

I. Conservation of Energy

  • Definition: The total amount of energy in a system remains constant.
  • Transformation: Energy can change forms (kinetic to potential, kinetic to heat), but the total energy is conserved.
  • Principle: A powerful principle used to solve problems that are difficult with kinematics or dynamics alone.
  • Method for Solving Problems:
    1. Determine initial energies: gravitational potential, elastic potential, and kinetic energies.
      • Write a mathematical expression including each of these energies.
      • Include energy from work being done.
    2. Determine final energies: gravitational potential, elastic potential, and kinetic energies.
      • Write a mathematical expression including each of these energies.
      • Include energy from work being done (Friction is a form of work).
    3. Apply the principle of conservation of energy: Initial total energy = Final total energy.
    4. Solve for the requested value.

II. Conservation of Energy - Isolated Systems

  • Isolated System: A system where no energy or matter can enter or leave.
  • Context: Total mechanical energy is conserved (no energy losses due to friction or heat).
  • Example 1:
    • A 50 kg object falls 490.5 m. What is the speed of the object just before impact with the ground?
  • Example 2:
    • A snowmobile driver with a mass of 100 kg traveling at 50 m/s slams into a snow drift. If the driver sinks 0.50 m into the snow drift before stopping, what is the retarding force applied by the snow drift?

III. Conservation of Energy - Non-Isolated Systems

  • Non-Isolated System: Mechanical energy can be "lost" as heat energy due to friction.
  • Conservation: The total amount of energy is still conserved, but some mechanical energy converts to thermal (heat) energy.
  • Example 5:
    • A 5.0 gram bullet enters a wooden block at 350 m/s and exits the 20 cm wide block at 150 m/s. What was the force applied to the bullet by the block?
  • Example 6:
    • Consider an object sliding down an inclined plane. A 25 kg object resting at the top of a 15 m high inclined plane begins to slide down the plane. At the bottom of the plane the object has a speed of 14.0 m/s.
      • A. How much heat energy was produced?
      • B. If the incline is 35.0 m long, what is the frictional force?

IV. Practice Problems

  1. A motorcycle rider is trying to leap across the canyon as shown in the figure by driving horizontally off the cliff. When it leaves the cliff, the cycle has a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side. (46.2 m/s)
  2. A 6.00 m rope is tied to a tree limb and used as a swing. A person starts from rest with the rope held in a horizontal orientation, as in the figure. Ignoring friction and air resistance, determine how fast the person is moving at the lowest point on the circular arc of the swing. (10.8 m/s)
  3. One of the fastest roller coasters (2000 kg) in the world is the Magnum XL - 200 at Cedar Point Park in Sandusky, Ohio. This ride includes an initial vertical drop of 59.3 m. Assume that the roller coaster has a speed of nearly zero as it crests the top of the hill.
    • A. If the track was frictionless, find the speed of the roller coaster at the bottom of the hill. (34.1 m/s)
    • B. The actual speed of the roller coaster at the bottom is 32.2 m/s. If the length of track is 125 m, what is the average frictional force acting on the roller coaster? (1.01x1031.01 x 10^3 N)

V. Hand-in Assignment

  1. An 80.0 kg box is pushed up a frictionless incline as shown in the diagram. How much work is done on the box in moving it to the top? (Hint, think energy, not forces.) (5.49 kJ)
  2. A 75 g arrow is fired horizontally. The bow string exerts an average force of 65 N on the arrow over a distance of 0.90 m. With what speed does the arrow leave the bow string? (39 m/s)
  3. In the high jump, the kinetic energy of an athlete is transformed into gravitational potential energy. With what minimum speed must the athlete leave the ground in order to lift his center of mass 2.10 m and cross the bar with a speed of 0.80 m/s? (6.5 m/s)
  4. A 50.0 kg pole vaulter running at 10.0 m/s vaults over the bar. Assuming that the vaulter's horizontal component of velocity over the bar is 1.00 m/s and disregarding air resistance, how high was the jump? (5.05 m)
  5. If a 4.00 kg board skidding across the floor with an initial speed of 5.50 m/s comes to rest, how much thermal energy is produced? (60.5 J)
  6. A roller coaster is shown in the drawing. Assuming no friction, calculate the speed at points B, C, D, assuming it has a speed of 1.80 m/s at point A. (24.3 m/s, 10.1 m/s, 18.9 m/s)
  7. A water skier lets go of the tow rope upon leaving the end of a jump ramp at a speed of 14.0 m/s. As the drawing indicates, the skier has a speed of 13.0 m/s at the highest point of the jump. Ignoring air resistance, determine the skier’s height H above the top of the ramp at the highest point. (1.38 m)
  8. A roller coaster vehicle with occupants has a mass of 2.9x1032.9 x 10^3 kg. It starts at point A with a speed of 14 m/s and slides down the track through a vertical distance of 25 m to B. It then climbs in the direction of point C which is 36 m above B. An interesting feature of this roller coaster is that due to cost-over-runs and poor planning, the track ends at point C. The occupant is the chief design engineer of the roller coaster ride. Estimate the speed of the vehicle at point B and then determine whether the fellow survives the ride. (26 m/s)
  9. The speed of a hockey puck (mass = 100.0 g) decreases from 45.00 m/s to 42.68 m/s in coasting 16.00 m across the ice.
    • a. How much thermal energy was produced? (10.17 J)
    • b. What frictional force was acting on the puck? (0.6357 N)
  10. During an automobile accident investigation, a police officer measured the skid marks left by a car (mass = 1500 kg) to be 65 m long. If the frictional force on the car was 7.66 kN during the skid, was the car going faster than the 100 km/h speed limit before applying the brakes? (slower)
  11. A 45.0 kg box initially at rest slides from the top of a 12.5 m long incline. The incline is 5.0 m high at the top. If the box reaches the bottom of the incline at a speed of 5.0 m/s, what is the force of friction on the box along the incline? (1.3x1021.3 x 10^2 N)
  12. For the pulley system illustrated to the right, when the masses are released, what is the final speed of the 12 kg mass just before it hits the floor? (7.0 m/s)

Hot Wheels Activity

  • Problem: What is the relationship between the potential energy of a car at the top of a hill and its kinetic energy at the bottom of a hill? How much heat was lost due to friction? What is the average frictional force of the track on the car? (Hint: You cannot use conservation of energy to calculate the speed at the bottom of the ramp. Why?)
  • ramp h (measure)
  • dx (measure) dy (measure)
  • Ep=mghE_p = m g h
  • Ek=½mv2E_k = ½ m v^2
  • table top