牛頓運動定律：練習題

A constant resultant force of \( 12 \text{ N} \) acts on an object of mass \( 3.0 \text{ kg} \) for a duration of \( 5.0 \text{ s} \). According to Newton’s laws of motion, what is the change in velocity of the object during this time interval?

A block of mass \( m \) is sliding down a rough slope inclined at an angle \( \theta \) to the horizontal. The block moves at a constant velocity. Which expression correctly defines the magnitude of the frictional force acting on the block in terms of the acceleration of free fall \( g \)?

A block of mass \( M \) is pulled along a horizontal surface by a constant force \( F \) applied at an angle \( \theta \) above the horizontal. If the coefficient of friction between the block and the surface is \( \mu \), which expression correctly represents the acceleration \( a \) of the block in terms of the acceleration of free fall \( g \)?

A constant resultant force of \( 25 \text{ N} \) acts on an object of mass \( 5.0 \text{ kg} \) for a duration of \( 4.0 \text{ s} \). What is the change in the momentum of the object during this time interval?

Two blocks, X and Y, with masses \( m_X \) and \( m_Y \) respectively, are connected by a light inextensible string passing over a smooth pulley. Block X sits on a smooth horizontal table while block Y hangs vertically. When the system is released from rest, the acceleration is found to be \( a \). If the mass of block Y is doubled, the new acceleration is \( 1.5a \). What is the ratio of the masses \( \frac{m_X}{m_Y} \)?

A student states that if no net force acts on an object, it must be stationary. Use Newton’s first law of motion to explain why this statement is incorrect.

A ball of mass \( m \) moving with velocity \( u \) strikes a wall and rebounds with velocity \( v \) in the opposite direction. If the contact time with the wall is \( \Delta t \), derive an expression for the average net force exerted by the wall on the ball in terms of these variables.

A high-speed elevator of mass \( 1500 \text{ kg} \) is moving upwards. At a certain instant, the tension in the cable is \( 18.5 \text{ kN} \) and the elevator is experiencing a constant air resistance (drag force) of \( 450 \text{ N} \). Using Newton's second law of motion, calculate the magnitude and direction of the elevator's acceleration at this instant. (Take \( g = 9.81 \text{ m s}^{-2} \))

A spacecraft of mass \( 1200 \text{ kg} \) is moving in deep space at a constant velocity of \( 450 \text{ m s}^{-1} \). To adjust its trajectory, a small thruster is fired, exerting a constant force of \( 300 \text{ N} \) at an angle of \( 60^{\circ} \) to its original direction of motion for a duration of \( 15 \text{ s} \). Friction and gravitational effects are negligible.

(a) Calculate the magnitude of the impulse provided by the thruster and state its direction relative to the original path.

(b) Using Newton's second law in terms of momentum, determine the change in the component of velocity parallel to the original direction of motion.

(c) Calculate the final speed of the spacecraft after the thruster has finished firing.

An object of mass \( m \) is initially at rest on a horizontal surface. A constant horizontal force \( F \) is applied to the object for a time interval \( \Delta t \). During this time, the object experiences a constant frictional force \( f \). After time \( \Delta t \), the force \( F \) is removed, and the object eventually comes to rest due to friction.

(a) State Newton’s second law in terms of momentum.

(b) Show that the maximum velocity \( v_{\text{max}} \) reached by the object is given by \( v_{\text{max}} = \frac{(F - f)\Delta t}{m} \).

(c) A second mass of \( 2m \) is placed on top of the first object. The same force \( F \) is applied for the same duration \( \Delta t \). Assuming the coefficient of friction remains constant (so the new frictional force is \( 3f \)), derive an expression for the ratio of the new maximum momentum to the original maximum momentum in terms of \( F \) and \( f \).

(d) Explain, using Newton’s third law, the relationship between the force the object exerts on the surface and the force the surface exerts on the object.