Crashtest Simulator

BEFORE DOING ANYTHING ELSE, MAKE THIS WINDOW AS WIDE AS POSSIBLE

The Fjord company is testing out its new model, the Ferraro. The blue Ferraro on the right is governed by the acceleration function you input on the left. Your inputs are the coefficients A, B, C, and D in the equation

a(t) = A t³ + B t² + C t + D

where a(t) is the acceleration at time t. The orange wall is 250 feet to the right of the starting position of the Ferraro which always starts with zero velocity. On the graph, the acceleration function is graphed in red, velocity is in yellow, and position in green.

You may also input the number of seconds to run the simulator below the graph. The speed at which the simulation runs is governed by the slider to the right.

The radio buttons on the bottom together with the redraw button will put colored dots on the graphs as targets. The same colored graph should pass through the center of the colored dot. If a multicolored dot appears, then the same colored graphs should pass through its center. CLICK HERE TO OPEN A WINDOW WITH THE CRASH TEST APPLET

Try to answer the following questions (questions 3 through 5 are quite challenging):

  1. What constant acceleration a(t) = D gets the Ferraro to the wall in ten seconds? What is the velocity at the wall?
  2. What linear acceleration a(t) = C t + D gets the Ferraro to the wall in ten seconds with velocity at the wall equal to zero?
  3. What quadratic acceleration a(t) = B t² + C t + D gets the Ferraro to the wall in ten seconds, providing a(0) = 0, v(10) = 0?
  4. What quadratic acceleration a(t) = B t² + C t + D gets the Ferraro to the wall in ten seconds, providing a(10) = 0, v(10) = 0?
  5. What cubic acceleration a(t) = A t³ + B t² + C t + D gets the Ferraro to the wall in ten seconds, under the constraints a(0)=a(10)=v(10)=0?

In order to acquaint you with the buttons, boxes and graphics of the simulator, this link takes you to a window giving a step-by-step walkthrough for question 4.

When answering questions 1 through 5, you should think about the relationships between acceleration, velocity, and position (which ones are derivatives of which other ones). You should also think about how the constraints influence your answers. Finally, on the trickier questions, it may help to consider the velocity or position functions as a guide to constructing the desired acceleration functions.

If you're having difficulties, follow this link for a brief explanation on how to attack such questions.