H3 Physics - HCI 2024 Newtonian Mechanics Tutorial with Solutions

Hwa Chong Institution (College) MOE H3 Physics 2024 1 H3 Physics 9814 - Newtonian Mechanics Problems 1 G.C.E. 'A' Level Physics Special Paper 2000 Question 7(c) Raindrops drip from rest from the edge of a roof at a height h above the ground. The drops leave the roof one after the other at a steady rate of n per unit time, where n is large. In the calculations which follow, neglect air resistance. (i) Find an expression, in terms of n, h and the acceleration of free fall g, for the number of drops in the air at any one instant of time. [2] (ii) Draw a sketch showing the approximate positions of the raindrops at the instant in (i). [1] (iii) Find, in terms of h, the height hmed above and below which equal numbers of raindrops are found, (hmed : the median height). [4] (iv) Find in terms of h, the average height <h> of the raindrops above the ground. [3] [ Hint: 12 + 22 + 32 + … + N2 = (N/6)(N+1)(2N+1)  N3/3 for large N.] 2 G.C.E. 'A' Level Physics Special Paper 2001 Question 1 A student, standing on the platform at Keppel Road Stat ion, Singapore, is watching the arrival of a train from Kuala Lumpur: She notes that the engine and the first carriage take 2.0 s to pass her, and the next two carriages take 2.4 s. The engine, and each of the carriages, is 20 m long. The braking deceleration of the train is constant. When the train comes to a stop, the student finds that the rear part of the last carriage is opposite her. (a) Calculate the deceleration of the train. [3] (b) Deduce the total number of vehicles (the engine plus the carriages) in the train. [3]

Hwa Chong Institution (College) MOE H3 Physics 2024 2 3 G.C.E. 'A' Level Physics Special Paper 2006 Question 10 A small object of density 900 kg m -3 is dropped into a very deep tank that contains water of density 1000 kg m-3. The object enters the water with a velocity of -7.00 m s-1, where the upwards direction is positive. (a) Assume that the object is not subject to a drag force as it moves through the water. (i) Calculate the maximum depth reached by the object and the total time from entering the water before it resurfaces. The hydrostatic upthrust experienced by a body of volume V when immersed in a fluid of density  is Vg, where g is the acceleration of free fall. [6] (ii) Sketch a full -page graph to show how the velocity v of the object depends on time t. Start at the instant the object enters the water and finish when it resurfaces. Mark the axes with important values of v and t. Label this graph A. Check your answer in (i) by considering an appropriate area on your graph. [3] (b) In fact the object resurfaced in exactly 11 s. This should be shorter than your answer in (a)(i). The discrepancy suggests that a drag force acts on the object. Assume that the drag force Fd is proportional to the velocity v of the object through the water. It is convenient to write this proportionality as Fd = -kmv where k is a constant and m is the mass of the object, which is also a constant. (Because the object under consideration is of fixed size, there is no need to consider the variation of drag force with the dimensions of the object. Similarly, because the object is falling through water only, there is no variation of drag force with viscosity.) (i) Obtain the equation of motion of the object in terms of acceleration dv dt , k, g and appropriate number(s). [2] (ii) Hence show that an equation relating the value of v to time t ( ) ( ) 1.09ln 1.09 7.00 −=− + kvkt k where k is measured in s-1, t in s and v in m s-1. [3] [Hint: ( ) ( )1 lndx a bxa bx b =− −− ] (iii) Solution of the equation in (ii) and substitution of the appropriate boundary conditions gives the value k = 0.092 s. (You are not asked to make this deduction.) Use the value of k to find the times taken for the object, 1. to travel from the surface to the lowest point, [2] 2. to return from the lowest point to the surface. [1] (iv) On the same axes as the graph in (a)(ii), sketch the graph to show how v depends on t when there is this drag force. Again, mark the axes with important values. Label this graph B. Comment on any features of areas on this graph compared with areas in the graph of (a)(ii). [3]

Hwa Chong Institution (College) MOE H3 Physics 2024 3 4 G.C.E. 'A' Level Physics Special Paper 1999 Question 6 A particle of mass m strikes a horizontal plane with velocity u at an angle θ to the plane. The collision is not perfectly elastic, and after the impact the particle moves off with velocity v at an angle ϕ to the plane as shown in Fig. 6.1. In such a collision, the component of velocity parallel to the plane remains constant, but the magnitude of the component normal to the plane after the impact is e times its magnitude before, where e is a constant less than one. (i) Find the fraction F of the kinetic energy of the particle lost in the impact. Give your answer in terms of e and θ. [5] (ii) Describe the main features of a collision, similar to the one shown in Fig. 6.1, but in which the value of the constant e is exactly one. [2] (b) A ball bounces inelastically down a flight of steps in a plane perpendicular to the front edges of the steps as shown in Fig. 6.2. Each step is 0.20 m high and 0.30 m deep. It is observed that the ball always bounces exactly in the middle of each step, and that after each bounce it rises to the height of the previous step. Air resistance can be neglected. (i) Show that the value of the constant e for these impacts ( i.e., the vertical component of velocity immediately after impact divided by the vertical component of velocity immediately before) is 0.71. [2] (ii) Show also that the horizontal component of velocity of the ball is 0.62 m s-1. [4]

Hwa Chong Institution (College) MOE H3 Physics 2024 4 (c) At the bottom of the steps in (b) the ball continues to bounce along a horizontal pavement as shown in Fig. 6.3. The collisions of the ball with the pavement have the same of e (0.71) as for the impacts with the steps. Show that the ball does bounce beyond a distance from the bottoms of the steps and calculate smax.