December 2015 Exam Review July-15-14 10:39 AM Here are solutions to the December 2014 final exam. 1. [5 marks] A soccer ball is kicked from the ground so that it is projected at an initial angle of 39 degrees relative to the ground and with an initial speed of 28 m/s. The ground is flat and horizontal, and there is no air resistance. (a) Determine the time that the soccer ball stays in the air until it hits the ground. (b) Determine how far from the kicking point the ball hits the ground. (c) Determine the ball's maximum height above the ground. Solution: Exam Review Page 1
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2. [5 marks] The Earth orbits the Sun once per year (3.16 10 7 s) in an approximately circular orbit of radius 1.50 10 11 m. With respect to the Sun, determine (a) the Earth's angular speed. (b) the Earth's tangential speed. (c) the magnitude of the Earth's centripetal acceleration. Solution: Exam Review Page 3
3. [5 marks] A 40-kg child slides down a curved slide starting at the top with an initial speed of 4 m/s. (a) Suppose that the slide is frictionless. Determine the speed of the child when she is 3 m below her starting height. (b) Suppose that the slide has friction. Determine the speed of the child when she is 5 m below her starting height if she starts from rest and 300 J of thermal energy is dissipated during the motion. Solution: The first choice in solving such problems is to use the principle of conservation of energy. If faced with such a problem, the first thing you should ask yourself is whether there is any reason that you can't use the principle of conservation of energy. For example, if friction is present, so that work is done but you can't account for the energy change, then you may not be able to use the principle of conservation of energy. In Part (a), there is no friction, so use energy conservation. In Part (b), friction is present, but we are told how much work friction does, so we can still use the principle of conservation of energy. Exam Review Page 4
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4. [5 marks] Two pucks collide on an air-hockey table, where friction between the pucks and the table can be assumed to be zero. Puck A has a mass of 0.025 kg and initially moves East with a speed of 5.5 m/s. Puck B has a mass of 0.050 kg and is initially at rest. After the pucks collide, Puck A moves at an angle of 65 degrees North of East and Puck B moves at an angle of 37 degrees South of East. Determine the speeds of the two pucks just after the collision. Exam Review Page 6
Solution: Use the principle of conservation of momentum. Remember that momentum is a vector, and therefore has two components. Also remember that each component of the total momentum of the system is conserved. Also note that nowhere in the statement of the problem is the word "elastic" mentioned, so we can't assume that kinetic energy is conserved. Exam Review Page 7
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5. [4 marks] Two banked curves have the same radius. Curve A is banked at an angle of 13 degrees and Curve B is banked at an angle of 19 degrees, where both angles are measured relative to the horizontal. A car can travel around Curve A without relying on friction at a speed of 18 m/s. Determine the speed at which the same car can travel around Curve B without relying on friction. Solution: Draw a free-body diagram! The diagram below shows a vertical cross section, so that the car (represented by the dot) is coming straight out of the page at you, moving in a horizontal circle, with its acceleration pointing straight to the left, towards the centre of the circle. Because the acceleration is towards the left, it makes sense to place the coordinate system so that one axis points to the left and the other points perpendicular to it. Exam Review Page 9
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6. [4 marks] In the figure, the two strings pass over massless, frictionless pulleys. The coefficient of friction between the 80-kg block and the table top is 0.1. When the 80-kg block is released, all three blocks begin to move. (a) Determine the blocks' acceleration. (b) Determine the tension in each string. Solution: Draw a free-body diagram for each block. Choose a coordinate system for each block in a compatible way (i.e., make sure the positive directions are compatible) so that you can use just one single symbol for the acceleration of each block. Remember that the accelerations of the three blocks are the same, but you will have to use three separate symbols for them if you don't choose your positive directions wisely. Then write Newton's second law for each direction for each block, and solve the resulting system of equations. Exam Review Page 12
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Now back-substitute to determine the tensions in the strings: 7. [4 marks] A bowling ball rolls along a rack without slipping. At the bottom of the slope the ball's translational speed is 3.50 m/s. Exam Review Page 14
Determine the ball's translational speed after it reaches the top of the slope, which is 0.76 m above the bottom of the slope. Friction can be ignored and the ball's moment of inertia is (2/5)MR 2, where R is the radius of the ball. Solution: See the other file. 8. [4 marks] A 10-g bullet is fired into a 2.5-kg wooden block that is attached to one end of a massless spring that has stiffness constant 845 N/m. The other end of the spring is fixed in place, and the spring is initially unstretched. The block rests on a horizontal, frictionless surface. Once the bullet strikes the block it quickly stops and becomes embedded into the block, and the block + bullet system oscillates with an amplitude of 0.2 m. Determine the initial speed of the bullet. Solution: Exam Review Page 15
For the full solution see the other file. 9. [4 marks] A subatomic particle called a pion has mass 2.41 10 28 kg. A short time after being formed (the pion's lifetime) the pion transforms into other particles. A pion moves at 99% of the speed of light relative to a laboratory. An observer who is stationary relative to the laboratory measures the lifetime of the pion as 3.5 10 8 s. (a) Determine the pion's lifetime as measured by a hypothetical observer moving along with the pion. (b) According to the hypothetical observer in Part (a), how far does the laboratory move during the pion's lifetime? (c) Determine the pion's total momentum and total energy as measured by an observer stationary relative to the laboratory. Solution: We'll ignore this one, as we didn't include the chapter on special relativity in our course this semester. 10. [10 marks] Circle the best response in each case. (a) You stand on a bathroom scale in an elevator, when the elevator suddenly begins accelerating upwards. The scale reading i. increases somewhat. ii. decreases somewhat. iii. does not change. iv. decreases suddenly to zero. v. [There is not enough information.] (b) You simultaneously release two balls from a height; one you drop straight down, and the other you throw horizontally. Which ball reaches the ground first? i. The ball dropped straight down reaches the ground first, because it travels a shorter distance. ii. The ball thrown horizontally reaches the ground first, because its speed is greater. Exam Review Page 16
iii. The two balls reach the ground at the same time, because their vertical motion is the same. iv. [It depends on the masses of the balls; the heavier ball falls faster.] v. [It depends on the masses of the balls; the heavier ball falls more slowly.] (c) Why is it difficult for a car to negotiate a curve at high speed? i. A huge force is pushing the car outward. ii. The sliding friction force is too large. iii. The faster the car moves, the harder it is for the driver to steer. iv. The magnitude of the friction force may not be large enough to provide the necessary acceleration. v. Because of Newton's third law, the car's reaction force may be too great. (d) How would the gravitational force exerted by the Sun on the Earth change if the distance between the Sun and the Earth suddenly shrank to half of its original distance, while the masses remained the same. The force would i. increase by a factor of 2. ii. increase by a factor of 4. iii. decrease by a factor of 2. iv. decrease by a factor of 4. v. remain unchanged. (e) Two satellites orbit the Earth in circular orbits. The radius of the orbit of Satellite A is smaller than the radius of the orbit of Satellite B. i. The speed of B is greater than the speed of A. ii. The period of A is greater than the period of B. iii. The speed of A is greater than the speed of B. iv. The angular speed of B is greater than the angular speed of A. v. [There is not enough information given.] (f) For an object moving in a circle at a constant speed, i. the net force on the object is zero, because the centripetal force exactly balances the centrifugal force. ii. the net force on the object is zero, because the angular acceleration is zero. iii. the net force on the object is directed towards the centre of the circle. iv. the net force on the object is directed away from the centre of the circle. v. the net force on the object is directed inwards, but not exactly towards the centre, because the object has a tangential velocity. Exam Review Page 17
(g) An object slides in a straight line on a horizontal frictionless surface. In order to continue its motion at a constant speed, i. a constant pushing force is needed. ii. a steadily increasing pushing force is needed. iii. a steadily decreasing pushing force is needed. iv. no external force is needed. v. no external force is needed only if the object contains its own force. (h) Object A has mass 10 kg and Object B has mass 2 kg. Object A collides with Object B and momentum is conserved in the collision. i. The force that object A exerts on Object B is greater than the force that Object B exerts on Object A. ii. The force that object A exerts on Object B is less than the force that Object B exerts on Object A. iii. The force that object A exerts on Object B is equal to the force that Object B exerts on Object A. iv. [It depends on whether the collision is elastic or inelastic.] (i) You pick up a book that is lying on the floor and place it on a desk. The total mechanical energy of the book i. has increased, because its potential energy has increased. ii. has decreased, because it is farther from the centre of the Earth. iii. remains the same, because it is still at rest. iv. remains the same, because no work is done on the book. v. remains the same, because the kinetic energy of the book does not change. (j) The laws of motion that form the core of this course are based on the work of i. Archaeopteryx of Jurassic. ii. Loki of Marvel. iii. Shaquille of Louisiana. iv. Monty of Python. v. Isaac Newton of Woolsthorpe. Exam Review Page 18