Lesson 6: How to Calculate Kinetic Energy

Transcription

1 KREUTTER:WORK AND ENERGY 1 Lesson 6: How to Calculate Kinetic Energy 6.1 Hypothesize (Derive a Mathematical Model) In a car crash testing facility, engineers evaluate the reaction of a car to a front impact. To create such an impact, a rod pushes a block of mass m on wheels over a distance d. This causes the block to accelerate from an initial to a final velocity. To measure the smashing potential of this block, let s determine the change in the block s kinetic energy after the rod pushes it a distance d. The initial and final states of the process are pictured to the right. vi d vf a) The block is your system; the rod is an external object. Draw a force diagram for the block. Use it to find an expression for the force that the rod exerts on the block in terms of its mass m and acceleration a. b) Use a kinematics equation to convert the acceleration a in the equation from part (a) into an expression involving the block s initial and final speeds v i and v f. Substitute this into the expression for force from part (a). c) Substitute the expression for force from part (b) into the expression for work when the force is parallel to the displacement, W = Fd, and then simplify. d) Using the given work-energy bar chart, develop a mathematical representation of this process in terms of work, initial kinetic energy, and final kinetic energy. Make sure to check the consistency of the graph with your system. Compare this expression to the one from part (c). + 0 before the block after the block is is lifted lifted K i + U g,i + U s,i + W = K f + U g,f + U s,f + U int - e) What characteristics of an object do you expect kinetic energy to depend on? Its mass? Velocity? Acceleration? Height?

2 f) By comparing your answers from parts (c) and (d), do you see a term that could represent kinetic energy and that depends on the characteristics that you think kinetic energy should depend on? g) Show that the units of this quantity are equal to the units for energy, joules. This change in kinetic energy of the block system is caused by the work done by the piston on the block. From that relationship, we find the accepted value for the kinetic energy of a system at a given moment is equal to 6.2 Test a Hypothesis 2 2 K 1 mv. Use a Hot Wheels car and launcher. Design an experiment to test the mathematical model you developed for kinetic energy (see the guidance below). Then design a second experiment that uses kinematics instead of work-energy. You should evaluate the consistency between the two methods. Include all of the elements below in a short lab report (one per group). Everyone s handwriting must be represented (i.e., each person in the group is responsible for at least one section). a) State clearly the hypothesis that you will test in the experiment. b) Play with launcher and the car and decide what features of their behavior you can explain using the concepts of kinetic energy. c) Think of experiments that you can perform whose outcomes you can predict using the ideas of kinetic and energy conservation. What other ideas/concepts can you use? Draw a picture. Decide what quantities you will measure and what quantities you will calculate. Decide what objects are in your system and whether any external objects do work on it. d) Make a prediction of the outcome of the experiment based on the idea being tested (the hypothesis). e) What are the additional assumptions that you are making? If these assumptions are not valid, how will they affect your result? f) Perform the experiment as many times as you think is necessary, collect the data, and calculate the result. How close is it to your prediction? g) What is another experiment that you can perform that does not use the ideas of energy to determine the same quantity? Perform it and analyze the results. Are the results of two experiments close within experimental uncertainty? (This is the second experiment that uses kinematics). h) What is your judgment about the hypothesis that you were testing?

3 KREUTTER:WORK AND ENERGY Regular Problem Kelly drives a truck which has twice as much mass as Heather s car. Kelly also drives twice as fast as Heather. Which statement is true about Kelly and her truck s kinetic energy (K) compared to that of Heather and her car? i. The truck has 4 times the K of the car. ii. All that can be said is that the truck has more K. iii. iv. The truck has twice the K of the car. The truck has 8 times the K of the car. Now that you have derived and tested mathematical expressions for both gravitational potential energy and kinetic energy, you can start solving problems. Below is the problem solving strategy that you might want to follow to learn how solve those problems as a physicist. Problem-Solving Strategy: Work-Energy Problems Sketch and Translate: Read the problem 3 times and visualize the situation/process. Sketch the physical process described in the problem. Include an initial state and a final state and a reference frame. Put all givens on the sketch. Make sure they are in consistent units. Decide on your system. Objects such as Earth, springs, and surfaces of interacting objects are usually included in the system. Objects that belong to the system do no work on each other but do possess different types of energy. External objects can do work on the system objects, thus causing the system s energy to change. Simplify and represent using the work-energy bar chart: Decide what internal or external interactions you can ignore. Construct a work-energy bar chart. Use the bars to represent the initial energies in the system, the work done on the system by any external objects, and the final energies in the system. Consider whether the following change: A system object s elevation above Earth (gravitational potential energy); A system object s speed (kinetic energy); An elastic system object (like a spring) stretches or compresses (elastic potential energy); The surface temperature of system objects increase as they rub against each other while one moves relative to the other (internal thermal energy change); A system object s shape changes during a collision (internal potential energy). Represent Mathematically: Apply the generalized work-energy principle; Convert the bars in the bar chart into a mathematical description of the process (one term for each bar in the bar chart). Solve and Evaluate: Use the mathematical description of the process to determine the unknown. Evaluate the results (units, magnitude, and limiting cases) to make sure they make intuitive sense.

4 6.4 If you drop a 0.3 kg baseball from a window 20 m above the ground, how fast will the ball be moving the instant before it hits the ground? Use the problem solving strategy. 6.5 If a stretched slingshot has 100 J of elastic potential energy, how fast will a 0.5 kg softball be moving right after the launcher fires it? Using energy representations, how high will the softball go? Use the problem solving strategy. 6.6 Reason Think back to the Newton s Cradle. Use your knowledge of energy to explain the simulation of this device: Specify the system, its initial and final states, and any assumptions you made.

5 KREUTTER:WORK AND ENERGY A crane lifts a 50-kg crate so that the crate s speed increases from 0 m/s to 5.0 m/s over a vertical distance of 10.0 m. Draw a bar chart representing this process. What is the force that the crane exerts on the crate? Use the problem solving strategy. Specify the system, its initial and final states, and any assumptions you made. Explain how these assumptions affect your answer. 6.8 A man throws a 0.4-kg softball upward at an initial speed of 10 m/s. How fast will it be traveling when it passes 1/3 of its maximum elevation? Use the problem solving strategy. 6.9 Two identical water balloon slingshots are stretched the same distance so that they both have the same elastic potential energy. The mass of one water balloon is 2/3 of the mass of the other water balloon. a) Which water balloon leaves the slingshot traveling at a faster speed? b) How much faster is this water balloon traveling?

6 6.10 Equation Jeopardy Write a problem and draw an energy bar chart that would require the mathematical equation below to solve it. Reflect: What did you learn about kinetic energy? How did you learn it? If you were to ask two questions about kinetic energy to find out if a person understands this concept, what would those questions be?