Position and movement (kinematics) ：練習問題

A particle moves along a straight line. Its position changes from \(x = -8 \text{ m}\) to \(x = +2 \text{ m}\). What is the displacement of the particle?

A car is moving with a velocity of \(12 \text{ m s}^{-1}\). It then accelerates uniformly over a distance of \(80 \text{ m}\) until its velocity reaches \(28 \text{ m s}^{-1}\). Calculate the acceleration of the car.

A projectile is launched from ground level with an initial velocity of magnitude \(u\) at an angle \(\theta\) above the horizontal. Air resistance is negligible, and the acceleration due to gravity is \(g\). If the time taken for the projectile to reach half of its maximum vertical height for the first time is \(T\), what is the maximum horizontal range \(R\) of the projectile?

A runner completes exactly three-quarters of a lap on a circular track with a radius of \( 40 \text{ m} \). What is the magnitude of the runner's displacement from the starting point?

A car starts from rest and accelerates uniformly along a straight road. If it covers a distance of 100 m in the first 5.0 seconds, what is the speed of the car at the end of these 5.0 seconds?

A displacement-time graph for an object moving in a straight line shows a straight line segment passing through the origin with a constant positive slope. Describe the motion of the object.

An object starts from rest and accelerates uniformly to a velocity of $$15 \text{ m s}^{-1}$$ in $$5.0 \text{ s}$$. It then continues to move at this constant velocity for another $$2.0 \text{ s}$$. Calculate the total displacement of the object during the entire $$7.0 \text{ s}$$ journey.

A train travels between two stations, P and Q, separated by a distance of \(1650 \text{ m}\). Starting from rest at P, the train accelerates uniformly at \(1.0 \text{ m s}^{-2}\) until it reaches its maximum velocity, \(V_{\text{max}}\,\). It then maintains this constant velocity for \(20.0 \text{ s}\). Finally, it decelerates uniformly at \(2.0 \text{ m s}^{-2}\) until it comes to rest exactly at Q. Calculate the maximum velocity \(V_{\text{max}}\) attained by the train during the journey, giving your answer to three significant figures.

A package is released from a hot air balloon when the balloon is at a height of 150 m above the ground and is ascending vertically at a constant velocity of 10.0 m s ^-1. Air resistance is negligible. Take the acceleration due to gravity as 9.8 m s ^-2 and assume the upward direction is positive.

(a) Calculate the time elapsed, measured from the moment of release, until the package reaches its maximum height.

(b) Determine the maximum height reached by the package above the ground.

(c) Calculate the total time taken from the moment of release until the package hits the ground.

(d) Determine the velocity of the package just before it hits the ground. Sketch the velocity-time graph for the package from the moment of release until it hits the ground. Label the initial velocity, time to reach maximum height, and final velocity on your sketch.