A car comes to a stop six seconds after the driver applies the brakes. While the brakes are on, the following velocities are recorded:
Time since
brakes applied
(sec) 0 2 4 6
Velocity
(ft/sec) 83 40 11 0
Give lower and upper estimates for the
distance the car traveled after the brakes were applied. An upper estimate =
A lower estimate =
To find the lower and upper estimates for the distance the car traveled after the brakes were applied, we need to calculate the area under the velocity-time graph.
First, let's plot the given data points on a graph, with time on the x-axis and velocity on the y-axis.
Time (sec) Velocity (ft/sec)
0 83
2 40
4 11
6 0
Now, to estimate the distance traveled, we need to calculate the area under the graph. Since the graph represents the velocity (ft/sec), the area under the curve will give us the distance traveled (in feet).
To calculate the lower estimate, we assume a lower value for velocity during each time interval. In this case, we can assume a constant velocity equal to the final velocity recorded (0 ft/sec) throughout each time interval. Therefore, the lower estimate for the distance is the sum of the areas of four rectangles:
Rectangle 1: Width = 2 sec, Height = 0 ft/sec
Rectangle 2: Width = 2 sec, Height = 0 ft/sec
Rectangle 3: Width = 2 sec, Height = 0 ft/sec
Rectangle 4: Width = 2 sec, Height = 0 ft/sec
To calculate the lower estimate, we add the areas of these rectangles:
Lower Estimate = (2 sec * 0 ft/sec) + (2 sec * 0 ft/sec) + (2 sec * 0 ft/sec) + (2 sec * 0 ft/sec)
= 0 ft + 0 ft + 0 ft + 0 ft
= 0 ft
Therefore, the lower estimate for the distance traveled by the car after the brakes were applied is 0 ft.
To calculate the upper estimate, we assume a higher value for velocity during each time interval. In this case, we can assume a constant velocity equal to the initial velocity recorded (83 ft/sec) throughout each time interval. Therefore, the upper estimate for the distance is the sum of the areas of four rectangles:
Rectangle 1: Width = 2 sec, Height = 83 ft/sec
Rectangle 2: Width = 2 sec, Height = 83 ft/sec
Rectangle 3: Width = 2 sec, Height = 83 ft/sec
Rectangle 4: Width = 2 sec, Height = 0 ft/sec
To calculate the upper estimate, we add the areas of these rectangles:
Upper Estimate = (2 sec * 83 ft/sec) + (2 sec * 83 ft/sec) + (2 sec * 83 ft/sec) + (2 sec * 0 ft/sec)
= 166 ft + 166 ft + 166 ft + 0 ft
= 498 ft
Therefore, the upper estimate for the distance traveled by the car after the brakes were applied is 498 ft.
To summarize:
Lower Estimate = 0 ft
Upper Estimate = 498 ft