Chapter 3: Applications of Differentiation
Section 3.10: Antiderivatives
A car makes a panic stop with a constant deceleration of −30 ft/s2 and leaves a skid mark of 180 ft. How fast was the car traveling when the brakes were first applied?
Note: This example is a variant of an exercise found on page 247 of a Stewart calculus text (ISBN 0-534-13212-X, 1991). It is included to show that often, applied problems require more "problem-solving skill" than actual calculus.
This author recalls learning more than 60 years ago when he was in Grade 6, the strategy of identifying "Given, Find, Solution" for word problems. It really does help when analyzing a problem to identify what information is known, what is to be discovered, and what relationships connect the known to the unknown.
Assume the car travels to the right on an x-axis, and first applies the brakes at x=0. Then, the motion of the car is described by xt=−30 t22+v0 t. (The calculus has been invoked by twice antidifferentiating x″t=−30, and using the conditions x′0=v0 and x0=0.)
The rest of the solution is algebraic. First, the total time taken to bring the car to a stop is found by solving xt=180 for
The initial velocity can now be found by solving the equation x′t^=0 for v0. For either value of t^ this equation becomes
from which it follows that v0=10800=603≐103.92 ft/s.
Converting this to miles per hour gives 603 ×36005280≐70.86 mph.
Converting this to kilometers per hour gives 114.033 km/h, where Maple's Units Calculator Assistant, as per Figure 3.10.5(a), was used.
Figure 3.10.5(a) Units Calculator Assistant (Tools≻Assistants) used to convert speed in ft/s to km/h
Tools≻Load Package: Student Calculus 1
Apply the AntiderivativePlot command to the acceleration to obtain the velocity function x′t
Apply the AntiderivativePlot command to the velocity to obtain the position function xt
Prefix the command with "xt="
Context Panel: Assign Function
→assign as function
Write the equation xt=180 and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻t
Context Panel: Conversions≻To List
Context Panel: Assign to a Name≻T
→solutions for t
→assign to a name
Form the equation x′t^=0 using either value of t^.
Context Panel: Solve≻Solve
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