Question:

The position-time graph of an object's motion is shown below.

a) Determine the average velocity of the object from t = 0 s to t = 5 s.

b) Determine the instantaneous velocity at t = 3 s.

c) Determine the object's displacement from t = 0 s to t = 4 s.

d) Is the object at rest at any time during this time interval? If yes, at what time(s)?

Answer:

a) The average velocity is calculated by dividing the displacement of the object by the time interval. On a position-time graph, displacement is given by the vertical distance between the initial and final positions.

At t = 0 s, the position of the object is 0 m, and at t = 5 s, the position is 30 m. Therefore, the displacement of the object is:

Displacement = final position - initial position = 30 m - 0 m = 30 m

The time interval is t = 5 s - t = 0 s = 5 s.

Average velocity = Displacement / Time interval = 30 m / 5 s = 6 m/s

Therefore, the average velocity of the object from t = 0 s to t = 5 s is 6 m/s.

b) Instantaneous velocity at a specific time is given by the slope of the tangent to the position-time graph at that particular time. To find the instantaneous velocity at t = 3 s, we draw a tangent line at that point.

The slope of the tangent line is given by Δy / Δx, where Δy represents the change in position and Δx represents the change in time.

From t = 2 s to t = 4 s, the position changes from 20 m to 30 m, and the time changes from 2 s to 4 s.

Δy = 30 m - 20 m = 10 m

Δx = 4 s - 2 s = 2 s

Instantaneous velocity at t = 3 s = Δy / Δx = 10 m / 2 s = 5 m/s

Therefore, the instantaneous velocity of the object at t = 3 s is 5 m/s.

c) Displacement is given by the area under the velocity-time graph. To find the displacement from t = 0 s to t = 4 s, we calculate the area of the triangle formed by the graph.

The base of the triangle is t = 4 s - t = 0 s = 4 s. The height of the triangle is the velocity at t = 4 s, which is 8 m/s.

Displacement = (1/2) * base * height = (1/2) * 4 s * 8 m/s = 16 m

Therefore, the object's displacement from t = 0 s to t = 4 s is 16 m.

d) An object is at rest when its velocity is zero. Looking at the graph, the object is at rest at t = 2 s and t = 5 s since the velocity is zero at these times.

Therefore, the object is at rest at t = 2 s and t = 5 s.