Unit 4: Projectiles ( Angled Projectiles ) When dealing with a projectile that is not launched/thrown perfectly horizontal, you must start by realizing that the initial velocity has two components: an x-component and a y-component. Thus, the first step is to resolve (or break up) the initial velocity into its component vectors. V 1 V 1y = V 1 sinq q V 1x = V 1 cosq v 1 q We can now analyze the motion in the same way that we analyzed the motion of a horizontal projectile, only now V 1y 0. We can immediately set up our x/y chart, and write what we know, as shown below. v 1 q It is important to always be aware of which two points you are working between. For horizontal projectiles, this was very easy since there were only 2 points (start and finish). For angled projectile problems, there are at least 3 critical points, as shown at the right. In this section, we only deal with points 1, 2, and 3. Let s look at a few example problems to see how angled projectiles are handled. Example: A football is kicked (on level ground) with a speed of 20 m/s at an angle of 30 o above the horizontal. How far will this football go, and how high will it reach at its apex? 1 2 3 4
Notice how I always made sure that I knew whether or not I was working between points 1 & 2 or between points 1 & 3. Also, notice that the time that the ball took to reach its maximum height was half the time it took to reach its landing point. And, by the way, the maximum distance that a projectile travels is known as its RANGE. Example: A ball is thrown (on level ground) with a velocity of 15 m/s at 60 o above the horizontal. Find the ball s impact velocity upon striking the ground. SIMPLE PROBLEM J Just like throw-up / come-down problems, as long as the projectile is launched and lands at the same height, its speed at point 3 is the same as the speed at point 1, its direction simply changes. Therefore, the impact velocity of the ball is 15 m/s at 60 o BELOW the horizontal. 15 m/s 60 60 15 m/s Example: Another football is thrown (on level ground) with an unknown speed at an angle of 30 o. If the football s range was 100 feet, find the initial speed of the ball.
Example: At what times will the football from the previous problem be 4 feet above the ground? And, how far away from the original throwing position will these times occur? Example: Using the previous problem, find the velocity of the ball when it is 4ft above the ground.
Example: A football is kicked on level ground at a speed of 20 m/s at an angle of 45 o. Track the football s position and velocity at t =1 and t = 2 seconds. Time(s) Dx=v x Dt Dy= ½ a y (Dt) 2 + v 1y Dt (x, y) 0 (20cos45)(0) = 0 m (0.5)(-9.8)(0) 2 + (20sin45)(0) = 0 m (0,0) 1 (20cos45)(1) = 14.142 m (0.5)(-9.8)(1) 2 + (20sin45)(1) = 9.242 m (14.1, 9.2) 2 (20cos45)(2) = 28.284 m (0.5)(-9.8)(2) 2 + (20sin45)(2) = 8.68 m (28.3, 8.7) Time(s) v 2x v 2y = v 1y + adt v 2 q 0 (20cos45) = 14.142 m/s (20sin45) + (-9.8)(0) = 14.142 m/s 20 m/s 45 o AH 1 14.142 m/s (20sin45) + (-9.8)(1) = 4.342 m/s 14.794 m/s 17.1 o AH 2 14.142 m/s (20sin45) + (-9.8)(2) = -5.458 m/s 15.159 m/s 21.1 o BH
ANGLED PROJECTILES: In-Class Examples A) A cannon fires a cannonball at an angle of 30 o above the horizontal. The ball leaves the cannon at 40 m/s. Find: a. the horizontal component of the initial velocity. b. the vertical component of the initial velocity. c. the maximum height reached. d. the total flight time (assuming that it is fired and lands at the same elevation). e. the range of the cannon. f. the impact velocity of the cannonball. B) How fast must a cannonball be shot out of the cannon if its angle of elevation is 20 o above the horizontal and it must hit a target 100 m away (at the same elevation as the end of the cannon)? How long will it take to hit the target? C) At what times will the cannonball from the previous problem be at a height of 3 m above the ground? D) A projectile is fired at a speed of 60 m/s and an angle of 30 o above the horizontal from the top of a 100 m tall building. Find: i) the horizontal distance away from the building that the projectile lands. ii) the total flight time of the projectile. E) A projectile is fired at a speed of 60 m/s and at an angle of 30 o below the horizontal from the top of a 100 m tall building. Find: i) the horizontal distance away from the building that the projectile lands. ii) the total flight time of the projectile. F) If a projectile is fired at a speed of 50 m/s at an angle of 45 o above the horizontal, track the projectile s.. i) position at t = 1, 2, & 3 seconds. ii) velocity at t = 1, 2, & 3 seconds.
G) A field goal kicker is kicking a field goal from 50 yards away (from the 10 ft tall goal post). The ball is kicked at an angle of 25 o and a speed of 70 mph. If the ball is kicked straight at the field goal post, will it clear bottom bar? If so (or if not), by how much would it (or would it not) clear? (remember: g = -32.2 ft/s 2, 1 yd = 3ft, 1 mi = 5,280 ft, 1 h = 3600 sec) H) Would the ball in the previous problem have been on its way up or on its way down when reached the field goal post? Logically explain how you can tell. I) A ball kicked off the top of a 30 m tall building. It is kicked at an angle of 20 o above the horizontal with a speed of 20 m/s. Find: i. the horizontal distance away from the building that the ball will land. ii. the time that the ball will take to reach the ground. iii. the maximum height off the ground attained by the ball. Projectile Motion (angled, on level-ground) HOMEWORK Problems A 1. A golf ball is hit from level ground at an angle of 60 o above the ground with an initial speed of 40 m/s. Determine the (a) horizontal range and maximum height of the ball. 2. A cannon ball is fired (assume from ground level) at a speed of 1200 m/s with an angle of 30 o above the ground. Determine a. the maximum height b. horizontal range of the cannon ball. 3. On level ground, a ball is thrown forward and upward. The ball is in the air 2 sec, and strikes the ground 30 m from the thrower. With what speed and at what angle was the ball thrown? Assume that the ball is thrown and caught at the same height. 4. A broad jumper takes off at an angle of 20 o above the horizontal and jumps 0.60 m high. (This is D y max ) a. What is her forward velocity? b. How far does she jump? 5. A ball is fired with a velocity of 1700 m/s at an angle of 55 o above the horizon. Determine a. the ball s horizontal range b. the amount of time that the ball is in motion?
Projectile Motion (angled, or not) HOMEWORK Problems B 6. A ball is launched off the top of a 30 meter tall building with a speed of 20 m/s at an angle of 30 o above the horizontal. Find the ball s a. max height reached (above the ground). b. total flight time. c. max range. d. impact velocity. 7. A ball is launched off the top of a 30 meter tall building with an unknown speed at an angle of 40 o above the horizontal. The ball reaches a maximum height of 50 meters above the ground. Find the ball s a. initial speed. b. total flight time. c. landing distance away from the building. 8. A ball is rolled off the building from problem #28 above. It rolls off horizontally at an unknown speed and lands 15 meters from the building. Find its initial speed and its impact velocity. 9. A tank fires a large bullet at a speed of 100 m/s. It leaves the barrel at an angle of 40 o. There is a very dense fog that is hovering 20 m above the ground. Find the times (after firing) that the bullet will enter and exit the fog layer. 10. A baseball player hits a fly ball to a height of 50.0 m. After the bat strikes the ball, how much time does a fielder have to get into position to make the catch? Projectile Motion (angled, and field-goals) HOMEWORK Problems C 11. A ball is thrown off of a 100 m cliff with a velocity of 25 m/s at an angle of 60 0 above the horizon. What is the ball s a. max height b. horizontal range c. impact velocity? 12. A golf ball is hit with an initial angle of 34 o with the horizontal and lands exactly 240 m down range on level ground. a. Find the initial speed of the ball b. Find the maximum height 13. A pilot cuts loose two of his fuel tanks in an effort to gain altitude. At the time of release, he was 120 m above the ground and traveling upward at an angle of 30 o above the horizontal, with a speed of 84 m/sec. For how long were the tanks in the air? 14. A daredevil is shot out of a cannon at 45 o to the horizon with an initial speed of 25.0 m/s. A net is placed 50 m from the cannon. At what height above the cannon should the net be placed in order to catch the daredevil? 15. A place-kicker must kick a football from a point 36.0 meters from the field goal post. The ball must clear the crossbar, which is 3.05 meters high. When kicked, the ball leaves at 20.0 m/s at an angle of 53 o above the horizon. By how much does the ball clear the bar? Was the ball on the way up or down when it went through?