2. 2 and 3 DIMENSIONAL MOTION Constant acceleration

3. The trajectory of an object projected with an initial velocity at the angle above the horizontal with negligible air resistance. Projectile Motion

4. a. 2.5 s b. 5.0 s c. 10 s d. 20 s e. 40 s Example: A projectile is fired from a cannon at a 30-degree angle with the ground and an initial velocity of 100 m/s. Assuming no air resistance and g = 10 m/s 2, calculate the time it will spend in the air. x y θ = 30  v 0 = 100 m/s y final = 0

5. Example

6. Shoot the monkey (tranquilizer gun) A zookeeper shoots a tranquilizer dart to a monkey that hangs from a tree. He aims at the monkey and shoots a dart with an initial speed v 0. The monkey, startled by the gun, lets go immediately. Will the dart hit the monkey? A.Only if v 0 is large enough. B.Yes, regardless of the magnitude of v o. C.No, it misses the monkey. If there is no gravity, the dart hits the monkey… If there is gravity, the dart also hits the monkey!

7. If there is no gravity, the dart hits the monkey… Continued

8. If there is gravity, the dart also hits the monkey! Note, that it takes the same amount of time to hit the monkey as in the no gravity case! Continued

9. This might be easier to think about… For the bullet: For the monkey: Continued

10. Example: A ball is hit from a platform, 1.0 m above the ground, with an initial velocity of 36.5 m/s and at an angle of 30° above horizontal. A 3 m high fence is 113 m from the base of the platform. Neglect air resistance. a)How long after the hit does the ball reach the fence? b)Does the ball go over the fence? c)What is the speed of the ball as it hits the ground? Fence 1 m 3 m θ x y

11. Example (continued): An alternative (and waaaaay longer) approach for part (c):

12. Example1: A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first? 1. A 2. both at the same time 3. B 4. need more information tAtA tAtA Shell A spends 2t A in the air, where t A is the time it takes for v y to become zero: 0 = v 0Ay — gt A. Shell A goes higher v 0Ay > v 0By t A > t B The vertical part of the motion dictates the time a projectile spends in the air. Example2: Two snowballs have been thrown with initial speed v 0, at angles θ 1 = 67.5° and θ 2 = 22.5°. Find the ratio of their ranges, R 1 /R 2 and the ratio of their time in flight t 1 /t 2. From an example in the previous lecture: