Kinematics of motion in a straight line：練習問題

A particle starts from rest and moves in a straight line with a constant acceleration of \( 2.5 \text{ ms}^{-2} \). Calculate the distance travelled by the particle in the first 4 seconds of its motion.

Particle \( A \) starts from point \( X \) and moves towards point \( Y \) with a constant speed of \( 5 \text{ ms}^{-1} \). At the same time, particle \( B \) starts from point \( Y \) and moves towards \( X \) with a constant speed of \( 15 \text{ ms}^{-1} \). Given that the distance \( XY \) is 120 m, how long after they start will the particles meet?

The velocity of a particle moving in a straight line is given by \( v = 0.6t^2 - 0.04t^3 \) for \( 0 \le t \le 15 \). Determine the maximum velocity attained by the particle during this interval.

The displacement, \( s \) metres, of a particle from a fixed point \( O \) at time \( t \) seconds is given by \( s = 4t^2 - t \). Find the velocity of the particle when \( t = 2 \).

A particle moves along a straight line. The velocity-time graph for its motion is a triangle, starting at \( (0, 0) \), reaching a maximum velocity of \( 10 \text{ ms}^{-1} \) at time \( t = T \), and then returning to rest at time \( t = 20 \text{ s} \). If the total displacement is 100 m, find the value of \( T \) such that the acceleration is twice the magnitude of the deceleration.

A particle moves in a straight line such that its velocity \(v\text{ m/s}\) at time \(t\text{ s}\) is given by \(v = 6t^2 - 2t\). Find the displacement of the particle between \(t = 0\) and \(t = 2\).

A car accelerates uniformly from rest to a speed of \(20\text{ m/s}\) in \(10\text{ s}\), then continues at this constant speed for another \(20\text{ s}\). Calculate the total distance traveled during the entire motion.

A particle moves in a straight line such that its velocity \( v \text{ ms}^{-1} \) at time \( t \text{ seconds} \) is given by \( v = 3t^2 - 12t + 9 \) for \( t \ge 0 \).

(a) Find the acceleration of the particle at the instant when \( t = 3 \).
(b) Find the values of \( t \) for which the particle is instantaneously at rest.
(c) Calculate the total distance travelled by the particle in the time interval from \( t = 0 \) to \( t = 2 \).