Mechanics - A Level Applied Maths & Physics: really challenging questions

1. An object that falls off a tree travels 16/25 th of the height of the tree during the last second. Assuming that g = 10 m/s², find the total time taken for the fall and the height of the tree.
2. A balloon is steadily rising at 5 m/s. After 4 seconds of its motion, a nail from the balloon falls off. Sketch a displacement-time graph for the motion of the nail from the beginning of the rise of the balloon up until the former hits the ground. State your assumptions, if any.
3. A car starts at a constant speed of 20 m/s. After 2 seconds, a motorcyclist starts from rest and moves at an acceleration of 10 m/s². Find the time taken by the motorcyclist to achieve the speed of the car. Do they meet up at that point? Explain. If not, when and where will they meet up? Work out this question by a velocity-time graph.
4. The width of a river is 12 m. A swimmer who can swim at 4 m/s wants to get to the other bank and swims straight across the river. Find the time taken by the swimmer to get there, if the water flows at 3 m/s. How far has he really swum? If he fails in achieving his goal, how should he swim in order to get to the other banks as fast as he can. Calculate the minimum time that he takes.
5. Two masses, 5kg and 8kg, are attached to a string and dropped from a tall tower while the string was slack. Calculate the tension of the string. State your assumption, if any.
6. A spring balance is attached to the string that is suspended at the ceiling of a lift. To the bottom hook of the spring balance, is attached a 5kg object. Calculate the reading of the spring balance, when the lift moves upwards at an acceleration of 2 m/s².
7. Two motorcars start off with a certain time gap and move with the same acceleration. Two minutes after the start of the second car, the distance covered by it is 4/9 of the distance covered by the first car by this moment. How long after the departure of the first car does the second car start?
8. A man, who stands inside an open cart that moves at constant speed, throws a ball up. Draw a distance-time graph to represent the displacement of both man and the ball on the same grid.
9. Two masses, 5kg and 3kg, are connected to a string and placed over a pulley. The lighter mass is on a smooth surface that is inclined to the horizontal at 30°. The other mass is vertically hanging down. Find the force that acts on the pulley, when the objects are in motion.
10. A parachutist, when comes down, bends his knees down just before reaching the ground. Explain this.
11. The mass of a gun and a bullet are 5 kg and 10 g respectively. Find the recoiling speed of the gun, when the bullet is fired at 200 m/s. If the shooter’s hands went backwards for 0.6 seconds, find the force exerted by him to keep the gun in balance.
12. If the bullet was aimed at a sack of sand, and it penetrated 4 cm into it, find the resistance offered by sand against the bullet.
13. You are standing still on a rough floor. Is there friction between your feet and the ground? Explain.
14. A thick rope is placed on a rough table. When the hanging section is ⅝ of the rope, the rope just starts to move down. Calculate the coefficient of friction between the rope and the table.
15. While ascending, a rocket of mass 6000 kg loses fuel at 0.1 kg/s, leaving the Earth at 600 km/h. Find the thrust generated by the rocket in the first 5 seconds. Explain why you cannot use F = ma in this calculation.
16. A pulley is suspended from a spring balance attached to the ceiling of a lift moving with a downward acceleration of 0.2g. Two loads with a total mass of 48 kg are attached to the rope of the pulley. The loads move with an acceleration of 0.3g relative to the lift. Find the masses of the loads and the reading of the spring balance. The masses of the pulley and rope, as well as the friction at the pulley axis, should be neglected.
17. A stone falls from rest under gravity. Taking three different points of its dissent, show that the sum of kinetic and potential energy remains the same. State your assumptions.



18. In the scenario below, an object is released from rest. The path from A to B is smooth. The ball leaves B horizontally and hits the ground at C. Calculate the value of h.
19. An object is placed on a rough surface. Its inclination is gradually raised up until it tends to move down. If the inclination at this point is Θ, derive an expression for the coefficient of friction.
20. The mass of an object is 5 kg and is placed on the same surface of coefficient of friction 0.25. A horizontal force of P is applied on the object up until it is about to move. Find two possible values of P.