非均勻加速度的運動：練習題

A student uses a video analysis tool to track the motion of a small sphere falling from rest through a viscous liquid. The position of the sphere is recorded at regular time intervals. Which of the following statements correctly describes the velocity-time graph derived from this motion?

An object of mass \(0.10 \text{ kg}\) falls through a fluid. The drag force \(D\) acting on the object is proportional to the square of its speed \(v\), such that \(D = kv^2\). The object reaches a terminal velocity of \(20 \text{ m s}^{-1}\). What is the acceleration of the object when its speed is \(10 \text{ m s}^{-1}\)?

An object falls vertically through a fluid where the drag force \(D\) is proportional to the square of its speed \(v\), such that \(D = kv^2\). The object reaches a terminal velocity \(v_t\). If the mass of the object is increased by a factor of 4 while its shape and size remain unchanged, what is the new terminal velocity in terms of \(v_t\)?

A small sphere falling through a viscous liquid has a terminal velocity \(v_t\). If the drag force acting on the sphere is directly proportional to its speed \(v\), determine the acceleration of the sphere in terms of \(g\) at the instant its speed is \(0.25v_t\).

A ball bearing is released from rest in a tall cylinder of oil. Describe how its acceleration changes as it reaches its terminal velocity and explain this in terms of the resultant force acting on the ball.

An object of mass \( 0.12 \text{ kg} \) is released from rest at the surface of a tall column of glycerol. The upward upthrust \( U \) acting on the object is constant and has a value of \( 0.25 \text{ N} \). As the object falls, it experiences a drag force \( D \) given by the expression \( D = 0.85v \), where \( v \) is the instantaneous velocity of the object.

(a) State and explain how the acceleration of the object changes from the moment it is released until it reaches its terminal velocity. Use the concept of resultant force in your answer.
(b) Calculate the terminal velocity of the object in the glycerol.
(c) Calculate the magnitude of the acceleration of the object at the instant its velocity is \( 0.50 \text{ m s}^{-1} \).
(d) Describe the shape of the displacement–time graph for this object, specifically identifying how the gradient of the curve changes at \( t = 0 \) and as \( t \) becomes very large.