*The kinetic energy of an object is given by K=(1/2)mv2, where v is the speed of the object and m is the mass of the object. Thus, the skater's kinetic energy is greatest at the lowest point of the track, where the skater is moving the fastest. Part B Change the Energy vs. Position graph to display only potential energy. As the skater is skating back and forth, where does the skater have the most potential energy? *The gravitational potential energy of an object is given by U=mgy, where y is the object's height above the potential energy reference, which is currently the ground. Thus, the skater's potential energy is greatest at the locations where the skater turns to go back in the opposite direction, where the skater is the highest above the reference line. Notice that the skater's potential energy is greatest where the kinetic energy is the lowest, and vice versa. Part C Because we are ignoring friction, no thermal energy is generated and the total energy is the mechanical energy, the kinetic energy plus the potential energy: E=K+U. Display the total energy in the Energy vs. Position graph. As the skater is skating back and forth, which statement best describes the total energy? *The mechanical energy (kinetic plus potential) is conserved. (Since there is no friction, the mechanical energy is equal to the total energy.) When the kinetic energy is relatively small, the potential energy is relatively large, and vice versa. Part D Ignoring friction, the total energy of the skater is conserved. This means that the kinetic plus potential energy at one location, say E1=K1+U1, must be equal to the kinetic plus potential energy at a different location, say E2=K2+U2. This is the principle of conservation of energy and can be expressed as E1=E2. Since the energy is conserved, the change in the kinetic energy is equal to the negative of the change in the potential energy: K2_K1=_(U2_U1), or _K1=__U2. Select the Show Grid option. Then, pull the bottom of the track down such that it is 1 m above the ground (click and drag on the blue circle on the bottom of the track). Confirm that the mass of the skater is set to 75.0 kg (select Choose Skater to view mass) and that the acceleration of gravity is set at 9.81 N/kg . Clear all plots, place the skater on the track 7 m above the ground, and look at the resulting motion and the graph showing the energetics. Match the approximate numerical values on the left with the energy type categories on the right to complete the equations. Drag the appropriate numerical values to their respective targets. Part E Based on the previous question, which statement is true? *Because the total energy is conserved, the kinetic energy at the bottom of the hill plus the potential energy at the bottom of the hill must equal the initial potential energy (since the initial kinetic energy is zero): Kbottom+Ubottom=Uinitial. Solving for the kinetic energy, we get Kbottom=Uinitial_Ubottom, or Kbottom=5145 J_735 J=4410 J. More generally, the change in the kinetic energy is equal to the negative of the change in the potential energy. Part F If the skater started from rest 4 m above the ground (instead of 7m), what would be the kinetic energy at the bottom of the ramp (which is still 1 m above the ground)? Part G One common application of conservation of energy in mechanics is to determine the speed of an object. Although the simulation doesn't give the skater's speed, you can calculate it because the skater's kinetic energy is known at any location on the track. Consider again the case where the skater starts 7 m above the ground and skates down the track. What is the skater's speed when the skater is at the bottom of the track? Express your answer numerically in meters per second to two significant figures. Part H When the skater starts 7 m above the ground, how does the speed of the skater at the bottom of the track compare to the speed of the skater at the bottom when the skater starts 4 m above the ground? Part I Change the potential energy reference line to be 7 m above the ground (select the Potential Energy Reference option, and click and drag on the dashed blue horizontal line to the 7 m grid line). Place the skater on the track 7 m above the ground, and let the skater go. Part J At the bottom of the hill, how does the kinetic energy compare to the case when the potential energy reference was the ground and the skater was released 7m above the ground? *The kinetic energy at the bottom of the track is equal to the amount of potential energy lost in going from the initial position to the bottom. Even though the total energy and initial potential energy are different from when the reference was the ground, the skater still loses the same amount of potential energy in going from 7 m down to the bottom of the track. Thus, when applying conservation of energy, any potential energy reference can be used! Part K Click Tracks in the upper-left corner of the window, and select Double Well (Roller Coaster). Then, click and drag on the blue circles to stretch and/or bend the track to make it look like that shown below.