Question:

A car is driving along a straight road. The position versus time graph for the car is given below:

1. What is the initial velocity of the car?
2. What is the acceleration of the car?
3. What is the average velocity of the car during the first 6 seconds?
4. At which point(s) on the graph is the car momentarily at rest?
5. During which time interval(s) is the car moving with constant velocity?

Answer:

1. To determine the initial velocity of the car, we need to find the slope of the position-time graph at the initial time.

   • At time t = 0 seconds, the position of the car is x = 20 meters.
   • At time t = 2 seconds, the position of the car is x = 40 meters.

   Using the equation for the slope:

   velocity = (change in position) / (change in time)

   velocity = (40 m - 20 m) / (2 s - 0 s)

   velocity = 20 m / 2 s

   velocity = 10 m/s

2. To find the acceleration of the car, we need to analyze the shape of the position-time graph. The acceleration of the car can be determined by finding the slope of the velocity-time graph.

   • Between t = 0 s and t = 2 s, the velocity remains constant at 10 m/s. This is represented by a horizontal line segment on the graph.
   • Between t = 2 s and t = 5 s, the velocity decreases linearly from 10 m/s to 0 m/s. This is represented by a downward sloping line segment on the graph.
   • Between t = 5 s and t = 6 s, the velocity remains constant at 0 m/s. This is represented by a horizontal line segment at the x-axis on the graph.

   Since the acceleration is the slope of the velocity-time graph, we can deduce that the acceleration of the car is -5 m/s².

3. To find the average velocity of the car during the first 6 seconds, we need to divide the total displacement by the total time.

   • The total displacement is given by the difference in position between t = 0 s and t = 6 s: displacement = x_final - x_initial displacement = 0 m - 20 m displacement = -20 m

   The total time is 6 s.

   average velocity = displacement / total time

   average velocity = -20 m / 6 s

   average velocity = -3.33 m/s (rounded to two decimal places)

4. The car is momentarily at rest at any point on the graph where the velocity of the car is zero. From the graph, we can identify two such points:

   • At t = 2 s, the velocity of the car is 10 m/s. This is the moment when the car stops momentarily.
   • At t = 5 s, the velocity of the car is also zero, indicating another momentary stop.

   Therefore, the car is momentarily at rest at points B and C on the graph.

5. The car is moving with constant velocity during time intervals where the slope of the position-time graph is constant. From the graph, we can identify two such time intervals:

   • Between t = 0 s and t = 2 s, the slope is constant and positive (a horizontal line segment).
   • Between t = 5 s and t = 6 s, the slope is constant and zero (another horizontal line segment).

   Therefore, the car is moving with constant velocity during these two time intervals.