2.2 Derive a Mathematical Model In a car crash testing facility, engineers evaluate the reaction of...

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2.2 Derive a Mathematical Model In a car crash testing facility, engineers evaluate the reaction of a car to an impact on its front. To create such an impact, a rod pushes a 1000 kg block on wheels over a distance Ax. This causes the block to accelerate from an initial to a final velocity. Let's determine the change in the block's kinetic energy after the piston pushes it a distance x. The initial and final states of the process are pictured to the right. V Vf a) Write a mathematical equation representing this process of changing the kinetic energy. It may help to create a work-energy bar chart for the process first. b) Write an expression for the work the piston does on the block during its displacement Ax. Substitute this into the expression in part a. c) Draw a force diagram for the block. Use it to find an expression for the force that the piston exerts on the block in terms of its mass m and acceleration a. Substitute this expression into the expression in part b. d) Use a kinematics equation to convert the acceleration a in the equation from part c into an expression involving the block's initial and final speeds v; and ve, and its displacement. Substitute this into the expression for force from part c. e) Examine the expression that you derived in part d. Do you see that the work that the piston did on the block equals the change in a quantity? Discuss how this expression can be used to write an expression for the kinetic energy of the block system. What characteristics of an object would you expect kinetic energy to depend on? Does the expression from part e seem like a term that could represent kinetic energy? g) Show that the units of this quantity are equal to the units for energy, joules. 2.2 Derive a Mathematical Model In a car crash testing facility, engineers evaluate the reaction of a car to an impact on its front. To create such an impact, a rod pushes a 1000 kg block on wheels over a distance Ax. This causes the block to accelerate from an initial to a final velocity. Let's determine the change in the block's kinetic energy after the piston pushes it a distance x. The initial and final states of the process are pictured to the right. V Vf a) Write a mathematical equation representing this process of changing the kinetic energy. It may help to create a work-energy bar chart for the process first. b) Write an expression for the work the piston does on the block during its displacement Ax. Substitute this into the expression in part a. c) Draw a force diagram for the block. Use it to find an expression for the force that the piston exerts on the block in terms of its mass m and acceleration a. Substitute this expression into the expression in part b. d) Use a kinematics equation to convert the acceleration a in the equation from part c into an expression involving the block's initial and final speeds v; and ve, and its displacement. Substitute this into the expression for force from part c. e) Examine the expression that you derived in part d. Do you see that the work that the piston did on the block equals the change in a quantity? Discuss how this expression can be used to write an expression for the kinetic energy of the block system. What characteristics of an object would you expect kinetic energy to depend on? Does the expression from part e seem like a term that could represent kinetic energy? g) Show that the units of this quantity are equal to the units for energy, joules.