Projectile Motion

Projectile Motion The motion of objects moving in 2 dimensions under the influence of gravity.

A running start… A long jumper is an example of a projectile. When the long jumper gets a running start, they have an initial horizontal velocity. Once he jumps off the ground, he now has a horizontal AND vertical velocity. We can break his motion up into components and use the kinematic equations to analyze the motion one direction at a time!

Constant acceleration problems (we can use kinematic equations now!) A hummingbird is flying in such a way that it is initially moving vertically with a speed of 4.6 m/s and accelerating horizontally at 11 m/s2. Assuming the bird’s acceleration remains constant for the time interval of interest, find (a) the horizontal and vertical distances through which it moves in 0.55 s and (b) its x and y velocity components at t = 0.55 s.

Projectiles follow a parabolic path! It would be a PERFECT parabola if there were no air resistance. Since there is air resistance, we don’t typically see perfect parabolic paths in our world. For the purpose of this class, we will be neglecting air resistance, giving us a perfect parabolic path when talking about projectiles (this will make the problems easier…)! Also, we will assume all horizontal velocity is CONSTANT, so we can use the (average) constant velocity equation: 𝑣 𝑥 = ∆𝑥 ∆𝑡

Projectiles launched horizontally Zero launch angle

Which will hit the ground first? 2 metal balls start at the same height. One is shot to the right from that height and the other is just dropped from the same height.

Why did they hit at the same time? If we talk about its components… The ball that was dropped had no initial horizontal velocity, but the ball that was shot did have an initial HORIZONTAL velocity. HOWEVER… what was the initial VERTICAL velocity of each ball?

Using the kinematic equations... When using the kinematic equations, if you are talking about vertical motion, you use Δy in place of the usual Δx’s. Also you would consider the v0 to be v0y. If you are doing horizontal motion, your velocity is the horizontal component (vx). (Since horizontal velocity is constant, initial and final are the same).

Projectiles launched horizontally The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45.0 m. Find the speed at which the rock was kicked.

Dropping a ball A person skateboarding with a constant speed of 1.30 m/s releases a ball from a height of 1.25 m above the ground. Given that x0 = 0 and y0 = h = 1.25 m, find x and y for (a) t = 0.250 s and (b) t = 0.500 s. (c) Find the velocity, speed, and direction of motion of the ball at t = 0.500 s. Think about… does the ball have an initially velocity in the x direction? Let’s look at a demo.

If a stunt man jumps from a 30. 0 m building to a pool that is 5 If a stunt man jumps from a 30.0 m building to a pool that is 5.0 m away from the building, with what initial horizontal velocity must the person jump?

A soccer ball is kicked horizontally off a 22 A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

Jumping a crevasse A mountain climber encounters a crevasse in an ice field. The opposite side of the crevasse is 2.75 m lower, and is separated horizontally by a distance of 4.10 m. To cross the crevasse, the climber gets a running start and jumps in the horizontal direction. (a) What is the minimum speed needed by the climber to safely cross the crevasse? (b) If, instead, the climber’s speed is 6.00 m/s, where does the climber land, and (c) what is the climber’s speed on landing?

Compare splashdown speeds Two youngsters dive off an overhang into a lake. Diver 1 drops straight down, diver 2 runs off the cliff with an initial horizontal speed v0. Is the splashdown speed of diver 2 (a) greater than, (b) less than, or (c) equal to the splashdown speed of diver 1?

Projectiles launched at an angle By neglecting air resistance, you can assume that the time and distance to the center of its motion (peak) will be the same on the way back down. Therefore if you find the time it takes to reach the peak, multiply it by 2 to find the total time. Likewise, the horizontal displacement to the peak can be multiplied by 2 to find the total horizontal displacement.

A rough shot Chipping from the rough, a golfer sends the ball over a 3.00-m-high tree that is 14.0 m away. The ball lands at the same level from which it was struck after traveling a horizontal distance of 17.8 m – on the green, of course. (a) If the ball left the club 54 degrees above the horizontal and landed on the green 2.24 seconds later, what was its initial speed? (b) How high was the ball when it passed over the tree?

CHALLENGE A trained dolphin leaps from the water with an initial speed of 12.0 m/s. It jumps directly toward a ball held by the trainer a horizontal distance of 5.50 m away and a vertical distance of 4.10 m above the water. If the trainer releases the ball the instant the dolphin leaves the water, will it hit the ball or miss it? Hypothesize with your group before attempting!!!

Monkey Time A zookeeper finds an escaped monkey hanging from a light pole. Aiming her banana gun at the monkey, the zookeeper kneels 10.0 m from the light pole, which is 5.00 m high. The tip of her gun is 1.00 m above the ground. At the moment the monkey releases the light pole, the zookeeper shoots. If the banana travels at 30.0 m/s, will the banana hit the monkey, go above it, or below it? http://www.physicsclassroom.com/mmedia/v ectors/mzi.cfm

An elevated green A golfer hits a ball from the origin with an initial speed of 30.0 m/s at an angle of 50.0 degrees above the horizontal. The ball lands on a green that is 5.00 m above the level where the ball was struck. (a) How long was the ball in the air? (b) How far has the ball traveled in the horizontal direction when it lands? (c) What are the speed and direction of motion of the ball just before it lands?

What a shot! The archerfish hunts by dislodging an unsuspecting insect from its resting place with a stream of water expelled from the fish’s mouth. Suppose the archerfish squirts water with an initial speed of 2.30 m/s at an angle of 19.5 degrees above the horizontal. When the stream of water reaches a beetle on a leaf at a height above the water’s surface, it is moving horizontally. (a) How much time does the beetle have to react? (b) What is the height of the beetle? (c) What is the horizontal distance between the fish and the beetle when the water is launched?

RANGE FORMULA and other beautifully symmetric things… This isn’t on your formula sheet, but can prove to be handy if you can remember it. Used to find the range of motion when a projectile has the same final and initial elevation. 𝑅= 𝑣 0 2 −𝑎 sin 2𝜃 At a given height the speed of a projectile is the same on the way up as on the way down. In addition, the angle of the velocity above the horizontal on the way up is the same as the angle below the horizontal on the way down.

Find the initial speed A football game begins with a kickoff in which the ball travels a horizontal distance of 45 yd and lands on the ground. If the ball was kicked at an angle of 40.0 degrees above the horizontal, what was its initial speed?

Dreaming of snow days… You and a friend stand on a snow-covered roof. You both throw snowballs with the same initial speed, but in different directions. You throw your snowball downward, at 40 degrees below the horizontal; your friend throws her snowball upward, at 40 degrees above the horizontal. When the snowball lands on the ground, is the speed of your snowball (a) greater than, (b) less than, or (c) the same as the speed of your friend’s snowball?

Let’s prove it! What angle do I need to set the projectile launcher at to aim straight at a ball held 2.5 meters off the ground and 1.5 meters away from the launcher?