A student is speeding down Highway 16 in her fancy red Porsche when her radar sy
ID: 1720565 • Letter: A
Question
A student is speeding down Highway 16 in her fancy red Porsche when her radar system warns her of an obstacle 400 feet ahead. She immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of her. The "black box" in the Porsche records the car's speed every two seconds, producing the following table. The speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a uniform rate.
Time since brakes applied (sec): 0, 2, 4, 6, 8, 10
Speed (ft/sec): 100, 85, 60, 25, 5, 0
A. What is your best estimate of the total distance the student's car traveled before coming to rest (note that the best estimate is probably not the over or under estimate that you can most easily find, use the trapezoidal approximation)?
distance =
B. Given the fact that the Porsche slows down during breaking, give a sharp
i. underestimate of the distance traveled: ans
ii. overestimate of the distance traveled: ans
C. Which one of the following statements can you justify from the information given?
A. The skunk was hit by the car.
B. The "black box" data is inconclusive. The skunk may or may not have been hit.
C. The car stopped before getting to the skunk.
Explanation / Answer
The whole point of differential calculus is to find the quantity of something at an instant in time. This is how/why Newton invented differential calculus.
Plot a graph of the speed versus time. If you have Excel use chart wizard to plot a scatter plot of speed on the y axis versus time on the x axis. Look at the slope of the lines between each point. That is an estimation of the acceleration (negative acceleration or "deceleration" in this problem)
Use the formula: s = Vot + ½at² to find the distance between each segment of line and then add them all up.
For the first one: Vo = 100ft/sec, t = 2 sec and a would be -12.5 ft/sec²
a from the graph is the slope (first derivative) of the line connecting the first two points so...
slope of line = m = Y2-Y1/X2-X1 = 85-100/2 = -15/2 = -12.5ft/sec² so a = -12.5 ft/sec²
Now you work the remaining 4 intervals and remember, to finally add up all of these distances to get the total distance.
In real life that black box would be taking several million readings a second and you'd have a smooth function that would be differentiable.
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