b. What is the x-distance from the foot of the cliff to the point of impact in the lake?

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1 PROJECTILE MOTION An object launched into space without motive power of its own is called a projectile. If we neglect air resistance, the only force acting on a projectile is its weight, which causes its path to deviate from a straight line. The projectile has a constant horizontal velocity and a vertical velocity that changes uniformly under the influence of gravity. HORIZONTAL PROJECTION If an object is projected horizontally, its motion can best be described by considering its horizontal and vertical motion separately. In the figure we can see that the vertical velocity and position increase with time as those of a free-falling body. Note that the horizontal distance increases linearly with time, indicating a constant horizontal velocity. 1. A cannonball is projected horizontally with an initial velocity of 10 m/s from the top of a cliff 50 m above a lake. a. In what time will it strike the water at the foot of the cliff? b. What is the x-distance from the foot of the cliff to the point of impact in the lake? c. What are the horizontal and vertical components of its final velocity? d. What is the final velocity at the point of impact and its direction?. A person standing on a cliff throws a stone with a horizontal velocity of 15.0 m/s and the stone hits the ground 47 m from the base of the cliff. How high is the cliff? 3. A box of supplies is dropped from an airplane that is located a vertical distance of 340 m above a lake. If the plane has a horizontal velocity of 70 m/s, what horizontal distance will the box travel before striking the water? (581 m) 1

2 4. A steel ball rolls off the edge of a table top 60 cm high. If the ball strikes the floor at a distance 50 cm from the base of the table, what was its velocity at the instant it left the table? (1.4 m/s) 5. Two tall buildings are 15 m apart. A ball is thrown horizontally from the roof of the first building 75 m from the ground. With what horizontal velocity must the ball be thrown if it is to enter a window of the second building 40 m from the ground? Draw a sketch first! (5.7 m/s) 6. A stone is thrown horizontally from the top of a building with an initial velocity of 00 m/s. At the same instant another stone is dropped from rest. a. Find the position and velocity of the second stone after 3 s. (44.1 m, 9.4 m/s) b. How far has the first stone traveled horizontally during the 3 s? (600 m) c. How far has it traveled vertically? (44.1 m) d. What are the horizontal and vertical components of its velocity after 3 s? (00 m/s, 9.4 m/s) 7. A bomber flying with a horizontal velocity of 500 km/h releases a bomb. Six seconds later the bomb strikes the ocean below. a. At what altitude was the plane flying? (176.4 m) b. How far did the bomb travel horizontally? (833.3 m) c. What are the magnitude and direction of its final velocity? (150.7 m/s, 3 )

3 PROJECTILE MOTION AT AN ANGLE The more general case of projectile motion occurs when the projectile is fired at an angle as shown in the figure. Problem solution: 1. Upward direction is positive. Acceleration (g) is downward thus negative.. Resolve the initial velocity vo into its x and y components: v v cos and v v sin 0x 0 0y 0 3. The horizontal and vertical components of its position at any instant is given by: 1 x v 0 xt and y v0 yt gt 4. The horizontal and vertical components of its velocity at any instant are given by: v x v 0 and v v gt x y 0 y 5. The final position and velocity can then be obtained from their components as a resultant. 3

4 8. An artillery shell is fired with an initial velocity of 100 m/s at an angle of 30 above the horizontal. Find: a. Its position and velocity after 8 s b. The time required to reach its maximum height and c. The horizontal distance (a.k.a. range R). 9. A baseball is thrown with an initial velocity of 10 m/s at an angle of 40 above the horizontal. How far from the throwing point will the baseball attain its original level? 10. A plastic ball that is released with a velocity of 15 m/s stays in the air for.0 s a. At what angle with respect to the horizontal was it released? b. What was the maximum height achieved by the ball? 11. a. Find the range of a gun which fires a shell with muzzle velocity vo at an angle θ. b. What is the angle at which the maximum range is possible? c. Find the angle of elevation θ of a gun that fires a shell with muzzle velocity of 10 m/s and hits a target on the same level but 1300 m distant. 1. A projectile is fired from the ground with an initial velocity of 96 m/s at an angle of 35. a. What will be the vertical and horizontal components of the initial velocity of this projectile? (78.6 m/s, 55 m/s) b. How long will it take this projectile to reach the highest point in its trajectory? (5.6 s) c. How long will this projectile be in the air? (11. s) d. What will be the velocity of this projectile at the highest point? (78.6 m/s) e. What will be the velocity of this projectile as it reaches the ground? (55 m/s) f. How high will this projectile be at the highest point of its trajectory? (154 m) g. What will be the range of this projectile? (885 m) 4

5 13. A projectile is launched at an angle of 30 with an initial velocity of 0 m/s. a. What is the highest point in its trajectory? (5.1 m) b. How much time is it in the air? (.04 s) c. What is its horizontal range? (35.3 m) 14. An arrow is shot into the air with a velocity of 10 m/s at an angle of 37 a. What are the horizontal and vertical components of its initial velocity? (95.8 m/s, 7. m/s) b. What is the position (x and y) after s? (19 m, 14.8 m) c. What are the magnitude and direction of the resultant velocity after s? (109. m/s, 8.7 ) 15. A shot put is released with a velocity of 1 m/s and stays in the air for.0 s. a. At what angle with the horizontal was it released? (55 ) b. What horizontal distance did it travel? (14 m) 16. An arrow was shot at an angle of 55º with respect to the horizontal. The arrow landed at a horizontal distance of 875 m. Find the velocity of the arrow at the top of its path. (55 m/s) 5