1 Non Linear Motion

2 Definitions: projectile - An object that is thrown,tossed, or launched. trajectory - The pathway of a projectile. Usually follows the shape of a parabola projectile motion - The motion of a projectile along a trajectory.

3 5 rules concerning projectile motion: 1. Projectiles always maintain a constant horizontal velocity, neglecting air resistance.

4 5 rules concerning projectile motion: 2. Projectiles always experience a constant vertical acceleration of 9.8 m/s2 downwards, neglecting air resistance.

5 5 rules concerning projectile motion: 3. Horizontal and vertical motion are completely independent of each other, therefore velocity of a projectile can be separated into horizontal and vertical components.

6 5 rules concerning projectile motion: 4. For a projectile beginning at the same height, the time it takes to reach its highest point equals the time it takes to fall from the highest point.

7 5 rules concerning projectile motion:continued 4. For a projectile beginning at the same height, the time it takes to reach its highest point equals the time it takes to fall from the highest point.

8 5 rules concerning projectile motion: 5. Objects dropped from a moving vehicle have the same velocity as the vehicle.

9 Horizontal & Vertical Speeds/Velocities of a Projectile

10 Horizontal & Vertical Distances of a Projectile

11 Classic Physics Question: A ball is dropped vertically, and one is launched horizontally, each from the same height. Which one hits the ground first? v Same time!

12 They both hit at the same time because they both fall the same distance. Although the thrown one has an additional horizontal velocity, that doesn’t affect its vertical motion. Breaking an objects motion into horizontal and vertical components is very useful!!

13 In the vertical direction, the ball experiences the usual acceleration from gravity. Notice its increasing falling distance. In the horizontal direction, the ball has a constant velocity. Thus, it covers regular, consistent horizontal distances.

14 Angled Projectiles: Projectiles aren’t always fired horizontally or vertically. They still can be broken into horizontal and vertical components: Notice how the horizontal component stays constant. Notice how the vertical component is affected by the acceleration from gravity

15 Equations: Horizontal Direction - (X) Vertical Direction - (Y) acceleration due to gravity (g) – -9.8 m/s 2 horizontal displacement - d h ==> v x = d x /t or d x = v x t vertical displacement - d v ==> (where v i is 0) d y = 1⁄2gt 2 horizontal velocity - v x = vcos  ; where v is the "muzzle" velocity or velocity of the toss as it leaves the hand vertical velocity - v y = vsin  where v is the "muzzle" velocity or velocity of the toss as it leaves the hand

16 Question: A sphere is dropped from a given height and a second sphere is fired from a launcher from the same height and at the same time as the first sphere is dropped. Compare the time it would take for the 2 spheres to hit the ground. Both spheres should land at the same time.

17 Question: *An object is dropped from an airplane prompting the following questions: *With what speed does the object leave the plane? The object leaves the plane with the same speed as the plane.

18 Question: *What type pathway does the object follow as it leaves the plane? The object follows a trajectory that is parabolic.

19 Question: *Where would the object be in relation to the plane when the object hits the ground? At the instant at which the object lands, it should be directly below the location of the plane.

20 Question: *What relationship does the vertical component of the motion have with gravity? dy = 1⁄2gt 2, which tells us that objects fall due to gravity, so the distance an object falls is dependent upon gravity and the time the object falls.

21 How Fast?, How Far? By thinking in terms of horizontal and vertical components, we can solve some simple problems: How fast did the boy throw the ball?

22 If the boy dropped one ball, and threw another horizontally, which would hit first? Tie We can use this fact to determine how long our projectile is in the air. Can you calculate how long it takes an object to fall 5m?

23 GIVEN: g = -9.8 m/s 2 d = -5 m v i = 0 m/s UNKNOWN: t- ? EQUATION : d y = 1/2 g t 2 SOLVE : d y = 1/2 g t 2 => t 2 = 1/2 g / d y SUBSTITUTION: t 2 = (1/2) 9.8 m/s 2 / 5m t = 1s Why was 5m used instead of 20m? Now that we know a time, we can use v =d/t since the horizontal velocity is constant. v = 20m / 1s = 20m/s

24 Range: Projectiles can be launched at any angle between 0 o and 90 o. This example leads to a long time in the air, but not much range. This example is more of a compromise of time in the air and range.

25 More on Range: Notice how two different angles can yield the same projectile range. What angle yields the maximum range possible?

26 Air Resistance: All the concepts previously mentioned are in the absence of air resistance (drag). The presence of air resistance changes the picture only slightly:

27 Into Orbit? If an object is throw horizontally at a very fast speed, it will actually go into orbit! Actually, the high speed just allows the object to fall around the earth instead of directly into it.

28 The velocity at which this occurs is 8km/s. This is equivalent to 18,000 mi/hr!!! When pieces of space junk fall to earth, their high speed causes them to burn up and disintegrate. Space craft fall or “orbit” around the earth at a higher altitude to avoid the friction from the atmosphere.

29 It would be incorrect to say there is no gravity as the space shuttle orbits the earth. In fact, there must be gravity so that the shuttle continuously falls “around” the earth. Otherwise they would fly off into space. The astronauts only feel weightless because the are continually in free fall.