Kinematics：練習問題

An object is dropped from rest from a height \(h\). At the same time, another object is thrown downwards with an initial velocity \(u\) from the same height. Which of the following statements is correct regarding their acceleration, assuming negligible air resistance?

A train accelerates from rest at a constant rate \(a_1\) for time \(t_1\) and then decelerates at a constant rate \(a_2\) for time \(t_2\) until it comes to rest. If the total distance travelled is \(S\), which expression represents the maximum velocity \(v_{max}\) reached by the train?

The displacement \(s\) of an object is plotted against time squared \(t^2\). The resulting graph is a straight line passing through the origin with a gradient of \(4.0\, \text{m s}^{-2}\). If the object started from rest, what is its acceleration?

The velocity \(v\) of a particle moving along a straight line is given by a velocity-time graph. The graph consists of a straight line from \((0, 0)\) to \((4, 12)\) followed by a horizontal line from \(t = 4\, \text{s}\) to \(t = 10\, \text{s}\). What is the average velocity of the particle over the interval \(t = 0\) to \(t = 10\, \text{s}\)?

An object moves along a straight line such that its displacement \( s \) varies with time \( t \) according to a non-linear relationship. Explain how a velocity–time graph can be used to determine the total displacement of the object over a specific time interval when the acceleration is not constant.

A displacement–time graph for a particle shows a curve where the gradient is initially positive and decreasing, becomes zero, and then becomes increasingly negative. Describe the motion of the particle in terms of its velocity and acceleration throughout this period.

A particle starts from rest and moves in a straight line with an acceleration \( a \) that increases linearly with time \( t \) such that \( a = kt \), where \( k \) is a constant. Derive an expression for the velocity \( v \) of the particle at time \( t \) and explain why the standard equations of motion (SUVAT) cannot be applied here.

The velocity-time graph for a car traveling along a straight road is shown. The car starts from rest at \( t = 0 \), accelerates uniformly to \( 20 \text{ m s}^{-1} \) in \( 10 \text{ s} \), maintains this constant speed for \( 30 \text{ s} \), and then decelerates uniformly to rest in a further \( 5 \text{ s} \).
(a) Sketch the velocity-time graph for this motion.
(b) Calculate the total displacement of the car.
(c) Determine the deceleration of the car during the final \( 5 \text{ s} \) of its journey.
(d) Sketch the corresponding acceleration-time graph for the car.

(a) Define acceleration and explain how it can be determined from a velocity-time graph.
(b) A sprinter starts a race from rest. For the first \( 3.0 \text{ s} \) of the race, her acceleration \( a \) varies with time \( t \) according to the expression \( a = 4.0 - 1.2t \).
(i) Calculate her instantaneous acceleration at \( t = 2.0 \text{ s} \).
(ii) Estimate the change in velocity of the sprinter between \( t = 0 \) and \( t = 3.0 \text{ s} \) by considering the area under an acceleration-time graph.
(iii) Explain why the equations of motion (SUVAT) cannot be applied directly to this sprinter's motion for the first \( 3.0 \text{ s} \).