Kinematics：練習問題

In the study of kinematics, a student plots a velocity–time graph for an object moving in a straight line. Which of the following quantities is determined by calculating the area under the graph?

A car is traveling along a straight road. The driver applies the brakes, and the car's velocity \( v \) varies with time \( t \) according to the equation \( v = 20 - 0.5t^2 \) for the interval \( 0 \le t \le 4.0 \text{ s} \). What is the instantaneous acceleration of the car at \( t = 3.0 \text{ s} \)?

A displacement–time graph for a toy car moving along a straight track is shown. The graph consists of a straight line with a positive gradient from time \( t = 0 \) to \( t = 5.0 \, \text{s} \), and then a horizontal line from \( t = 5.0 \, \text{s} \) to \( t = 10.0 \, \text{s} \). Which statement correctly describes the motion of the car during the interval from \( t = 5.0 \, \text{s} \) to \( t = 10.0 \, \text{s} \)?

Define average velocity and state the condition under which it would be equal to the instantaneous velocity of an object.

An object is moving with constant acceleration along a straight line. It passes point A with a velocity of \( 4.0 \, \text{m s}^{-1} \) and point B, which is \( 50.0 \, \text{m} \) further along, with a velocity of \( 16.0 \, \text{m s}^{-1} \). Calculate the acceleration of the object and the time taken to travel from A to B.

Describe how the velocity of an object is represented on a displacement-time graph and state what a horizontal line signifies.

An object moves along a straight line. Its displacement \( s \) from a fixed point is monitored over a period of \( 20.0 \, \text{s} \). For the first \( 8.0 \, \text{s} \), the object moves with a constant velocity of \( 5.0 \, \text{m s}^{-1} \). It then decelerates uniformly for the next \( 4.0 \, \text{s} \) until it is momentarily at rest. For the final \( 8.0 \, \text{s} \), it accelerates uniformly in the opposite direction at \( 1.2 \, \text{m s}^{-2} \).

(a) Calculate the displacement of the object at \( t = 8.0 \, \text{s} \).
(b) Determine the deceleration of the object between \( t = 8.0 \, \text{s} \) and \( t = 12.0 \, \text{s} \).
(c) Calculate the final velocity of the object at \( t = 20.0 \, \text{s} \).
(d) Calculate the total distance traveled by the object during the entire \( 20.0 \, \text{s} \) interval.

A remote-controlled car travels along a straight track. The motion of the car is described in three stages. In the first stage, it starts from rest and accelerates uniformly to a speed of \( 6.0 \, \text{m s}^{-1} \) in a time of \( 3.0 \, \text{s} \). In the second stage, it maintains this constant speed for \( 8.0 \, \text{s} \). In the final stage, the car decelerates at a constant rate, coming to rest in a further \( 4.0 \, \text{s} \).

(a) Calculate the total displacement of the car during the entire \( 15.0 \, \text{s} \) journey.
(b) Determine the magnitude of the acceleration during the first stage and the deceleration during the final stage.
(c) Calculate the average speed of the car for the whole journey.