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AP Physics 1 Exam Question
A student is conducting an experiment to analyze the motion of a trolley moving on a frictionless track. The student records the position of the trolley at different time intervals and plots a graph of position versus time.
The graph is shown below:
a) Based on the graph, determine the displacement of the trolley between t = 0 s and t = 6 s. Show your work and provide units for your answer.
b) At what time(s) is the velocity of the trolley equal to zero? Justify your answer using the graph.
c) Determine the average velocity of the trolley between t = 0 s and t = 8 s. Show your work and provide units for your answer.
Answer and Explanation
a) Displacement is the change in position of an object. In this graph, the displacement is given by the area under the curve between t = 0 s and t = 6 s. To calculate this, we need to find the area of the rectangle and the triangle individually, and then add them together.
- Rectangle Area: The rectangle has a width of 6 s (from t = 0 s to t = 6 s) and a height of 20 m. Therefore, the area of the rectangle is:
Rectangle Area = width * height = 6 s * 20 m = 120 m
- Triangle Area: The triangle has a base of 6 s and a height of 10 m (the change in position from t = 0 s to t = 6 s). Therefore, the area of the triangle is:
Triangle Area = 0.5 * base * height = 0.5 * 6 s * 10 m = 30 m
Total Displacement: The total displacement is the sum of the rectangle and triangle areas:
Total Displacement = Rectangle Area + Triangle Area = 120 m + 30 m = 150 m
Therefore, the displacement of the trolley between t = 0 s and t = 6 s is 150 meters.
b) The velocity of an object can be determined from a position versus time graph by finding the slope of the tangent line at each point. At the points where the tangent line is horizontal (slope = 0), the velocity of the object will be zero.
By analyzing the graph, we can see that the tangent line is horizontal (slope = 0) at two points: t = 2 s and t = 8 s.
At t = 2 s, the position of the trolley is approximately 40 m, and at t = 8 s, the position is approximately 100 m. At both these times, the velocity is zero as indicated by the horizontal tangent line.
c) Average velocity is defined as the total displacement divided by the total time interval. In this case, we are calculating the average velocity between t = 0 s and t = 8 s.
Total Displacement: From part a), we found that the total displacement between t = 0 s and t = 6 s is 150 m. We need to find the displacement between t = 0 s and t = 8 s.
The area under the curve from t = 6 s to t = 8 s is a rectangle with width 2 s and height -10 m (note the negative sign indicates the displacement is in the opposite direction). Therefore, the displacement between t = 6 s and t = 8 s is:
Rectangle Area = width * height = 2 s * -10 m = -20 m
Now we can calculate the total displacement:
Total Displacement = Displacement from t = 0 s to t = 6 s + Displacement from t = 6 s to t = 8 s = 150 m + -20 m = 130 m
Total Time Interval: The total time interval is t = 8 s - t = 0 s = 8 s.
Average Velocity: The average velocity is the total displacement divided by the total time interval:
Average Velocity = Total Displacement / Total Time Interval = 130 m / 8 s = 16.25 m/s
Therefore, the average velocity of the trolley between t = 0 s and t = 8 s is 16.25 m/s.