Linear motion：練習問題

A car travels along a straight road with constant acceleration. It passes two points, X and Y, that are separated by a distance of \( 100 \, \text{m} \). The time taken for the car to travel from X to Y is \( 5.0 \, \text{s} \). The speed of the car as it passes point Y is \( 25 \, \text{m s}^{-1} \). What is the acceleration of the car?

A car is travelling at a constant speed \( v \) on a straight level road. The driver has a fixed reaction time and applies a constant braking force to stop the car. The thinking distance is \( d_t \) and the braking distance is \( d_b \).
If the car were travelling at speed \( 2v \), how would the new thinking distance and braking distance compare to the original values?

A small ball is thrown vertically upwards in a uniform gravitational field with an initial velocity of \( 15.0\text{ m s}^{-1} \). If air resistance is negligible, what is the maximum vertical height reached by the ball above its point of release? (Take \( g = 9.81\text{ m s}^{-2} \))

A stone is thrown vertically upwards from the edge of a vertical cliff of height \( H \) with an initial speed \( u \). The stone eventually falls past the cliff edge and hits the ground at the base of the cliff after a total time \( T \). If air resistance is negligible, which equation correctly relates the height of the cliff to the other variables? (Use \( g \) for the acceleration of free fall).

A car is travelling at a constant speed of \( 20.0 \, \text{m s}^{-1} \) on a level road when the driver sees a hazard. The driver has a reaction time of \( 0.50 \, \text{s} \) before the brakes are applied. Once the brakes are engaged, the car decelerates at a constant rate of \( 5.0 \, \text{m s}^{-2} \) until it stops. What is the total stopping distance of the car?

Explain why the thinking distance for a driver is directly proportional to the initial speed of the vehicle, assuming a constant reaction time.

A car is travelling at a constant speed of \( 30.0 \, \text{m s}^{-1} \) when the driver suddenly sees a hazard \( 85.0 \, \text{m} \) directly ahead. Given the driver has a reaction time of \( 0.60 \, \text{s} \) and the car decelerates at a constant rate of \( 7.5 \, \text{m s}^{-2} \), calculate the total stopping distance and determine the speed at which the car would hit the hazard.

A car travels at velocity \(v\) before the driver applies the brakes with a reaction time \(t\). If the constant deceleration is \(a\), derive the expression for the total stopping distance \(d = vt + \frac{v^2}{2a}\) and state the thinking distance.

A small steel ball is released from rest at a height \( H \) above the ground. It falls vertically through a uniform gravitational field. The ball passes a tall window of vertical height \( 3.0 \, \text{m} \) in a time of \( 0.18 \, \text{s} \). Assume air resistance is negligible and use \( g = 9.81 \, \text{m s}^{-2} \).

(a) By considering the motion of the ball as it passes the window, calculate the velocity of the ball at the instant it reaches the top of the window.
(b) Calculate the height \( h \) from the point of release to the top of the window.
(c) A student intends to determine the acceleration of free fall \( g \) in the laboratory using a similar falling object and a pair of light gates. Explain how the student can use a graph of displacement \( s \) against time squared \( t^2 \) to determine \( g \), and suggest one way to minimize the percentage uncertainty in the measurements of time.

A driver is travelling at a constant speed of \( 32 \, \text{m s}^{-1} \) on a straight motorway. The driver notices a hazard ahead and applies the brakes. There is a reaction time (thinking time) of \( 0.60 \, \text{s} \) before the brakes are applied. Once the brakes are engaged, the car decelerates at a constant rate of \( 6.5 \, \text{m s}^{-2} \).

(a) Define thinking distance and braking distance.
(b) Calculate the total stopping distance for the vehicle from the moment the driver first sees the hazard until the car comes to rest.
(c) The motorway has a speed limit of \( 70 \, \text{mph} \) (approximately \( 31 \, \text{m s}^{-1} \)). A new safety system reduces the driver's reaction time to \( 0.20 \, \text{s} \). Calculate the percentage reduction in the total stopping distance caused by this system for a car travelling at \( 32 \, \text{m s}^{-1} \).
(d) Explain, without further calculation, why the braking distance of a car is proportional to the square of its initial speed.