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1 NT4E-QRT20: PROJECTILE MOTION FOR TWO ROCKS VELOCITY AND ACCELERATION GRAPHS II Two identical rocks are thrown horizontally from a cliff with Rock A having a greater velocity at the instant it is released than Rock B. For this exercise, you should ignore air resistance. Use a coordinate system with down as the positive vertical direction, away from the cliff as the positive horizontal direction, and with the origin at the bottom of the cliff directly below the release point, a) Sketch below the velocity vs. graphs for each of the rocks. Rock A (faster) Rock B (slower) b) Which rock hits the ground first? Both hit at the same. c) Which rock lands farthest from the base of the cliff? Rock A, the faster rock. Explain why you drew the graphs as you did and how you determined your answers. There are no forces in the horizontal direction (ignoring air resistance) so both rocks will have a constant horizontal velocity. Since Rock B is slower than Rock A, it will have a smaller horizontal velocity. Both rocks are thrown horizontally, so they have no initial vertical velocity. Both are acted on by gravity, so the slopes of their vertical velocity graphs are constant and equal to g, and both are positive because gravity is acting down and the positive direction is defined as down. Both rocks hit at the same but rock A hits farther from the cliff since it travels faster in the horizontal direction. ntipers Not for distribution 22

2 NT4E-QRT21: BASEBALL PROJECTILE MOTION VELOCITY AND ACCELERATION GRAPHS A baseball is thrown from point S in right field to home plate. The dashed line in the diagram shows the path of the ball. Home plate S For this exercise, use a coordinate system with up as the positive vertical direction and to the left as the positive horizontal direction, and with the origin at home plate. Select the graph from the choices below that best represents: 1) the horizontal velocity vs. graph A_ 2) the horizontal acceleration vs. graph C 3) the vertical velocity vs. graph I 4) the vertical acceleration vs. graph B A B C D E F G H I J K L M N O P None of these -Explain or Sketch graph Answer: There are no horizontal forces (neglecting air resistance) and the ball will have no horizontal acceleration, so the horizontal velocity will be constant. The positive direction is to the left, so the horizontal velocity will be positive. Since up is positive, the initial vertical velocity is positive, and the final vertical velocity is negative. The only vertical force is gravity acting downward, and so the acceleration is constant, negative, and equal in magnitude to g. Since the acceleration is constant, the slope of the velocity must be constant, so the vertical velocity must be a straight line sloping downward from its initial positive value to its final negative value. So the correct answers are (1) A; (2) C; (3) I; and (4) B. 23 ntipers

3 NT4E-CRT22: PROJECTILE MOTION FOR TWO ROCKS VELOCITY AND ACCELERATION GRAPHS II Two students throw two rocks horizontally from a cliff with +y different velocities. Both rocks hit the water below at the same +x but Rock B hits farther from the base of the cliff. For this exercise, you should ignore air resistance. Use coordinates where up is the positive vertical direction, away from the cliff is the positive horizontal direction, and the origin is at the top of the cliff at the point of release. Sketch below velocity vs. graphs for each rock. Rock A Rock B Rock A (closer) Rock B (farther) Explain how you determined your answer. Answer: There are no forces in the horizontal direction (ignoring air resistance) so both rocks will have a constant horizontal velocity. Since Rock B is lands farther than Rock A, it must have had a larger horizontal velocity. Both rocks are thrown horizontally, so they have no initial vertical velocity. Both are acted on by gravity, so the slopes of their vertical velocity graphs are constant and equal to g, and both are positive because gravity is acting down and the positive direction is defined as down. ntipers Not for distribution 24

4 NT4E-LMCT23: DROPPED PRACTICE BOMB LANDING POINT A military airplane is flying 1200 m above the ground at a speed of 200 m/s. It drops a practice bomb that hits the ground after traveling a horizontal distance of 3130 m. Ignore air resistance. 200 m/s 1200 m For each of the numbered changes below, use the lettered choices below to identify what will happen to the horizontal distance the bomb travels while falling compared to the situation above. (a) The horizontal distance will be greater than 3130 m. (b) The horizontal distance will be less than 3130 m but not to zero. (c) The horizontal distance will be equal to 3130 m. (d) The horizontal distance will be zero, i.e., the bomb will drop straight down. (e) We cannot determine how this change will affect the horizontal distance. For each of the following changes, only the feature(s) identified is(are) modified from the given situation above. 1) The plane s speed is tripled. A: The it takes to hit the ground is the same, but the bomb has a greater horizontal velocity. 2) The plane is climbing straight up at the release point. D: Since there is no horizontal velocity it will come straight back down if it misses the plane. 3) The plane is flying in level flight at an altitude of 1100 m. B: It will take less to reach the ground from a lower height, and the horizontal speed is the same. 4) The mass of the bomb is increased. C: The vertical acceleration is still g and is independent of the mass, so the path the bomb takes will 5) The bomb is thrown from the plane with a vertical downward velocity of 15 m/s. B: The acceleration of the bomb is the same, but with an initial downward vertical velocity it will 6) The plane is diving at a 20 angle and is at a height of 1200 m. B: Thontal velocity is smaller since only a component of the 200 m/s is horizontal. 7) The plane s speed decreases, and it is flying at an altitude of 1800 m. E: Without knowing how much the speed decreased (and doing a calculation) we cannot tell how these changes will affect the horizontal distance. Decreasing the speed acts to decrease the horizontal distance, while increasing the height acts to increase it. But we cannot tell which change will be most important. 25 ntipers

5 NT4E-LMCT24: CANNONBALL FIRED FROM AN ELEVATED CANNON MAXIMUM HEIGHT A cannonball is fired from a cannon located at the edge of a cliff. The muzzle of the cannon is 40 meters above the water. The cannonball has an initial horizontal velocity of v vertical = 45 m/s 80 meters per second and an initial upward vertical muzzle v velocity of 45 meters per second. horizontal = 80 m/s Identify, from choices (a) (d), how each change described below will affect the maximum height that the cannonball reaches above the water level. This change will: (a) increase the maximum height of the cannonball. (b) decrease the maximum height of the cannonball. 40 m (c) have no effect on the maximum height of the cannonball. (d) have an effect on the maximum height of the cannonball that cannot be determined. All of these modifications are changes to the initial situation that is shown in the drawing. 1) The cannon is adjusted so that the initial vertical velocity of the projectile is increased and the initial horizontal velocity is unchanged. Answer: (a) increase because the average speed while the cannonball is moving upward will increase. 2) The gun is adjusted so only the initial horizontal velocity of 150 m/s of the projectile is reduced and the initial vertical velocity is unchanged. Answer: (c) no effect since only the vertical velocity makes a difference in the height. 3) The cannon is moved to a lower cliff. Answer: (b) decrease since the cannonball starts from a lower point. 4) The mass of the cannonball is reduced without changing the initial velocities. Answer: (c) no effect since the gravitational force and mass are proportional so that the acceleration doesn t change. ntipers Not for distribution 26

6 NT4E-CT25: TOY TRUCKS ROLLING OFF TABLES WITH DIFFERENT HEIGHTS TIME Two toy trucks roll off the ends of tables. The heights of the tables, the speeds of the trucks, and the masses of the trucks are given. 8 m/s 300 g A 200 g 5 m/s B 50 cm 80 cm Will Truck A be in the air for a longer, a shorter, or the same as Truck B before it reaches the floor? Answer: a shorter because it has a smaller distance to fall. Both trucks are acted on by gravity, so they will both accelerate downward with the same acceleration g. Both trucks have no initial vertical velocity. The they are in the air depends only on initial height. Mass and horizontal velocity don t affect the the trucks are in the air. 27 ntipers

7 NT4E-CT26: TOY TRUCKS WITH DIFFERENT SPEEDS ROLLING OFF IDENTICAL TABLES TIME Two toy trucks roll off the ends of identical tables. The speeds and masses of the trucks are given. 8 m/s 500 g A 5 m/s 800 g B 30 cm 30 cm Will Truck A be in the air for a longer, a shorter, or the same as Truck B before it reaches the floor? Answer: the same because they have the same distance to fall. Both trucks are acted on by gravity, so they will both accelerate downward with the same acceleration g. Both trucks have no initial vertical velocity. The they are in the air depends only on initial height. Mass and horizontal velocity don t affect the the trucks are in the air. ntipers Not for distribution 28

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