1. A cannon shoots a clown directly upward with a speed of 20 m/s. What height will the clown reach?

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1 Physics R Date: 1. A cannon shoots a clown directly upward with a speed of 20 m/s. What height will the clown reach? How much time will the clown spend in the air? Projectile Motion 1:Horizontally Launched Projectiles Two dimensional motion is motion confined to a plane (x-y axis) Motion in a plane is broken down into two mutually perpendicular components. The component along the x-axis is referred to as horizontal motion while the component along the y-axis is referred to as vertical motion Horizontal motion is not affected by any external forces (as long as we consider ideal cases) Vertical motion is influenced by gravity *Mass does not matter* THE BIG IDEA What happens in the vertical dimension What happens in the horizontal dimension What happens in the x-dimension What happens in the y-dimension So What stays the same between the horizontal and vertical? Solving projectile motion problems The table: d vi a t x y For all projectile motion problems: a y = Vertical (y) motion is a x = Horizontal (x) motion is For horizontally (x) launched projectiles, the projectile is launched in the: direction A cannon fires a projectile horizontally (x) at 30 m/s. What is the horizontal (x) component of the initial velocity? What is the vertical (y) component of the initial velocity? For all horizontally (x) launched projectiles,

2 Horizontal (x) motion has no effect on vertical (y) motion. A gun is fired and a bullet is dropped simultaneously. Which bullet hits the ground first, the dropped or fired? For each part of each problem, determine whether you are solving a vertical or horizontal motion problem before solving. Label X or Y in the column to the left. 2. A cannon shoots a projectile horizontally (x) at 50 m/s from the top of a 70 m tall cliff. a) How much time does it take the projectile to reach the ground? b) What is the range (d x ) of the projectile? 3. A ball rolls toward the edge of a table that is 110 centimeters high and lands 2.0 meters away from its edge. a) How much time does it take the ball to reach the ground? b) Determine the speed with which the ball rolls of the edge. 4. A stunt car traveling toward the edge of a cliff at 25 meters per second lands 50 meters measured horizontally from the edge of the cliff. a) How much time does it take the car to reach the ground? b) Determine the height of the cliff. 2

3 5. A stuntman jumps off the edge of a 45 meter tall building to an air mattress that has been placed on the street below at 15 meters from the edge of the building. a) How much time does it take the stuntman to reach the ground? b) What minimum initial velocity does he need in order to make it onto the air mattress? Visualizing Horizontally Launched Projectiles A projectile is launched horizontally at 5 boxes per second. Create a motion/dot diagram along the top for the horizontal. Put an x on each spot Create a motion/dot diagram along the left for the vertical. Put a y on each spot Mark the intersections to see where the projectile would be in x-y space The path or trajectory of the projectile is a: Sketch acceleration vectors for each Sketch (x and y) velocity vectors for each 3

4 6. A car in a James Bond movie flies off the end of a cliff with a velocity of 20 m/s. If the car is in the air for 10 s, find the following information: a. The height of the cliff. b. The range the car traveled. c. The location of the car after 5 seconds in the x and y direction. 7. A 1 kg ball is rolled off the end of a 1 m high table with a velocity of 2 m/s. Find the following: a. The time the ball was in the air. b. The range the ball traveled. c. The impact velocity in the x and y direction AND the resultant velocity 4

5 Identifying Variables: Identify the following variables (i.e. d x d y v ix, a y, etc.) 8. A student standing on the roof of a 80-meter-high (A) building kicks a stone at a horizontal speed of 4 meters per second (B). How much time does it take the stone to hit the ground? What is the question asking for? What are the hidden or unstated variables? A: B: Fill in the table to the right 9. A ball is thrown horizontally at a speed of 14 meters per second (A) from the top of a cliff. If the ball hits the ground 4.0 seconds (B) later, approximately how high is the cliff? What is the question asking for? What are the hidden or unstated variables? A: B: Fill in the table to the right Horizontally Launched Projectile Practice What happens in the x-dimension stays in the x-dimension What happens in the y-dimension stays in the y-dimension What does this mean? Use the table at the right. A projectile is launched horizontally at 20 m/s. Four seconds later it hits the ground. What is the range of the projectile (d x or d y )? Which v i should I use to get the range? Which a? Solve for the range. x y d vi 20 m/s 0 a m/s2 t 4 s What is the initial height of the projectile (d x or d y )? Which v i should I use to get the range? Which a? Solve for the height. 10. A cannonball launches horizontally off a tall cliff at 40 m/s. Another ball is dropped. Which one will hit the ground first? Why? After 4 seconds, what is the range of the cannonball? If a train was traveling at 40 m/s, how far would that travel in 4 seconds? How does this compare to the cannonball? 5

6 11. The Enola Gay is traveling horizontally at 75 m/s at an altitude of 9470 m, when it drops the bomb. What is the initial horizontal and vertical velocity of the bomb? Create a motion/dot diagram along the top for the plane. Create a motion diagram for the bomb dropped underneath it. What does this mean for the plane and the bomb? Where is the plane when the bomb hits the ground? 12. A plane is flying horizontally at an altitude of 1200 m at a speed of 150 m/s. It drops a package to land on an island. How far away from the island should the plane be when it releases the package? (Hint: Create a table. You may need to find something in common between x and y) 13. Three cannons fire three different cannon balls horizontally off the top of the same 30-meter tall cliff different velocities as shown in the table below. Cannon Ball Velocity (m/s) A 40 m/s B 70 m/s C 90 m/s a) Which cannon ball hits the ground first? How do you know? b) What is the horizontal velocity of Cannon Ball A after 2 seconds? c) What is the horizontal component of Cannon Ball B s velocity when it hits the ground? d) How long does it take for Cannon Ball C to hit the ground? e) What is the horizontal range of Cannon Ball C? 6

7 14. A plane is flying horizontally at 120 m/s when it releases a package. The package crash lands on an island 12.7 seconds later. a) What is the horizontal component of the package s velocity the instant it hits the ground? b) What is the vertical component of v the instant it hits the ground? c) What is the resultant velocity the instant it hits the ground? Vector Components Into how many components can a vector be split? 15. A soccer ball is kicked at 25 degrees from the horizontal with an initial velocity of 25 m/s. a) What is the x-component of the ball s initial velocity? b) What is the y-component of the ball s initial velocity? 16. A little boy pulls a wagon with a force of 17 N at an angle of 23 degrees. a) What is the horizontal component of the boy s force? b) What is the vertical component of the boy s force? 3. Gravity pulls down on the box to the right with a force of 100 N. a) What is the component of the force parallel to the incline? b) What is the component of the force perpendicular to the incline? 7

8 Projectile Motion 2: Ground Launched Projectiles Ground launched projectiles are just like horizontally launched projectiles, except that v initial is not completely horizontal (v iy doesn t equal zero). 17. A cannon fires a shot with an initial velocity of 80 m/s at an angle 27 degrees above the horizontal a) What is the horizontal component of the initial velocity? (make a sketch) b) What is the vertical component of the initial velocity? c) At the peak of the trajectory, what is the horizontal component of the velocity? d) At the peak of the trajectory, what is the vertical component of the velocity? e) How much time does it take the projectile to reach the peak? (Make a table. Include v f ) 18. A baseball is hit with an initial velocity of 35 meters per second at an angle of 18 above horizontal. a) Calculate the vertical component of the ball s initial velocity b) How much time will it take the baseball to reach the top of its flight? c) How much total time will the baseball spend in the air? 19. A place-kicker strikes a football with a horizontal velocity of 22 meters per second and vertical velocity of 10 meters per second. The ball reaches its maximum height within 1.02 seconds. a) What is the total time that the ball will spend in the air? b) Calculate the horizontal distance that the football will travel during its flight. 20. A ball is thrown with an initial velocity of 15 meters per second at an angle of 20 above the horizontal. a) Determine the vertical component of the ball s initial velocity. b) Determine the ball s total time of flight. 8

9 Ground Launched Projectile Practice For any projectile motion, we know that: 21. A soccer ball is kicked with an initial velocity of 20 m/s at an angle of 30 degrees a) What is the horizontal component of the initial velocity? b) What is the vertical component of the initial velocity? Sketch acceleration along the path shown Sketch horizontal velocity along the path shown c) What is the horizontal component of velocity at the peak? Sketch vertical velocity along the path shown d) What is the vertical component of velocity at the peak? e) What is the horizontal component of velocity at the end? f) What is the vertical component of velocity at the end? g) How much time does it take the ball to reach the peak? h) How much total time does the ball spend in the air? i) What is the maximum height the ball reaches? j) What is the total range of the ball? 9

10 22. A baseball is hit with an initial velocity of 35 meters per second at an angle of 18 above horizontal. a. Sketch the baseball s trajectory. Indicate the point at which the ball s vertical velocity is 0. b. Calculate the vertical component of the ball s initial velocity. c. Calculate the time that it will take for the ball to reach the top of its flight. Explain how this amount of time is related to the total amount of time that the ball will remain in the air. d. Calculate the maximum height that the ball will reach. 23. A place-kicker strikes a football with a horizontal velocity of 22 meters per second and vertical velocity of 10 meters per second. The ball reaches its maximum height within 1.02 seconds. a. What is the total time that the ball will spend in the air? b. Calculate the horizontal distance that the football will travel during its flight. 24. A ball flies through a parabolic path and passes through points A, B, and C. Compare the a. vertical accelerations A, B, and C. b. vertical velocities at A, B, and C. c. horizontal accelerations A, B, and C. d. horizontal velocities at A, B, and C. 10

11 Ground Launched Projectiles: Angles Create a sketch showing a projectile launched at 0, 30, 45, 60, and 90 degrees. Label them as such. At what angle will the soccer ball have the greatest range? As the angle goes from 0-30 degrees, what happens to the range? At what angle will the soccer ball have the greatest height? As the angle goes from degrees, what happens to the height? At what angle will the soccer ball spend the most time in the air? As the angle goes from degrees, what happens to time in the air? 25. A bullet is fired horizontally and another bullet is dropped simultaneously from the same height. If air resistance is neglected, which bullet hits first, explain your answer. 26. Does a ball dropped out of a window of a moving car take longer to reach the ground than one dropped at the same height from a car at rest, explain. 27. A man is holding two rocks at equal heights. One is dropped and the other is thrown horizontally. (a) which rock hits first (b) which rock will have a greater velocity when it hits 28. A certain gun type fires bullets at 60 m/s regardless of its orientation. One Bullet is fired with the gun aimed at 10 above the horizontal, a second bullet is fired simultaneously at 45 above the horizontal and a third bullet is fired simultaneously at an angle of 60. Ignoring air resistance, answer the questions that follow and explain your choice for each one. (a) which hits the ground first, (b) which goes the highest, (c) which has the greatest range, (d) which has the largest x component (e) which has the greatest acceleration 11

12 Review Checklist Explain the assumptions needed to solve a horizontal projectile problem. 1. What can you assume about the horizontal and vertical accelerations of horizontally launched projectiles? 2. What critical assumption must you make about the initial vertical velocity of a horizontally launched projectile? 3. A cannon fires a cannonball horizontally from the top of a cliff with a speed of 80 m/s. What is the initial horizontal velocity of the cannonball? Sketch vectors for velocity or acceleration at different positions in a projectile s path. Compare velocities and accelerations at different points. 4. At each position: A, B, and C sketch a vector to represent the vertical and horizontal velocities of the object. b. Compare the vertical speed at A to the vertical speed at B. c. Compare the horizontal speed at B to the horizontal speed at C. d. Compare the vertical acceleration at A to the vertical acceleration at C. e. Compare the horizontal acceleration at A to the horizontal acceleration at B. Use equations to determine: time of flight; launch height; and/or horizontal distance for a horizontallylaunched projectile. 5. An object is thrown horizontally from a height of 10 meters with an initial velocity of 8.0 meters per second. How much time will it take before the object hits the ground? What horizontal distance did it travel? 6. A ball is thrown horizontally with an initial velocity of 5.0 meters per second. The ball travels 15 meters horizontally before it lands. How much time was the ball in the air for? What vertical height was it thrown from? 7. A projectile is fired horizontally from a height of 15.0 meters above the ground. The projectile travels 40 meters horizontally from the point from which it was fired before it lands. What initial velocity was the projectile fired with? Determine Vector Components, including components of a ground-launched projectile s initial velocity. 8. A projectile is launched with a velocity of 200 meters per second at an angle of 35 above the horizontal. What are the vertical and horizontal components of the projectile s initial velocity? Explain the assumptions needed to solve a ground-launched problem. 9. What can you assume about the horizontal and vertical accelerations of ground-launched projectiles? 10. What can you assume about the relationship between the time that it takes for a ground-launched projectile to reach its maximum height and its total time of flight? 12

13 Sketch vectors for velocity or acceleration at different positions in a projectile s path. Compare velocities and accelerations at different points. 11. At each position: A, B, and C sketch a vector to represent the vertical and horizontal velocities of the object. b. Compare the vertical speed at A to the vertical speed at B. c. Compare the horizontal speed at B to the horizontal speed at C. d.compare the vertical acceleration at A to the vertical acceleration at C. e. Compare the horizontal acceleration at A to the horizontal acceleration at B. Use equations to determine: time of flight; max height; horizontal distance; and/or launch angle for a ground-launched projectile. 12. A ball is thrown with an initial velocity of 15 meters per second at an angle of 20 above the horizontal. a. Determine the vertical component of the ball s initial velocity. b. Determine the ball s total time of flight. 13. A rock that is projected with an initial velocity of 24 meters per second at an angle of 60 above horizontal takes 2.12 seconds to reach its maximum height. a. What is the horizontal component of the ball s initial velocity? b. What total horizontal distance will the rock travel before it lands? 14. A projectile is fired with an initial vertical velocity of 12 meters per second and an initial horizontal velocity of 15 meters per second. a. Determine the maximum vertical height that the projectile will reach. b. Determine the total time the projectile will spend in the air. Relate launch angle to: time of flight; max height; and horizontal range 15. At what launch angle will a projectile have: a. Maximum height? b. Maximum range? c. Maximum time in air? 16. Compare a. the time spent in the air for a rock thrown at an angle of 30 to the time spent in the air for a rock thrown at 70. b. the max height reached by a rock thrown at an angle of 40 to the max height for a rock thrown at 10. c. the horizontal range for a rock is thrown at an angle of 40 to the horizontal range for a rock thrown at

14 Review Checklist Answers: 1. a x = 0, a y = -9.8 m/s 2 2. v iy = m/s 4. b. larger at B c. same d. same (-9.8 m/s 2 ) e. same (0) s, 11.43m 6. 3s, 44.1 m m 8. v iy = 115 m/s, v ix = 164 m/s 9. a x = 0, a y = -9.8 m/s t total = 2 x (t max height ) 11. b. less (zero) at B c. same d. same (-9.8 m/s 2 ) e. same (0) 12. a m/s b s 13. a. 12 m/s b m 14. a. 7.3 m b. 2.4 s 15. a. 90 b. 45 c a. Greater for 70 b. Greater for 40 c. Greater for 40 14