Projectile Motion

1. Projectile Motion • Concerns the flight of an object or body after it is free of support. (This includes objects that are dropped.) • The flight path of a projectile is called the trajectory. • Objects that are continuously being propelled (such as airplanes) aren’t considered projectiles.

2. Examples of Projectiles • Football • Javelin • Discus • Long jumper • Diver

3. A human in flight obeys the same projectile laws as any other object.

4. Factors Affecting the Trajectory of a Projectile • The relative height of projection • The angle of projection (the initial angle of the trajectory relative to horizontal) • The speed of projection (the velocity of the object when it is first released) • Air resistance and wind

5. speed of projection angle of projection height of projection Text page 336

6. Relative Projection Height • This is the release height compared to the final landing height of the projectile

7. Relative Projection Height • Relative projection height = 0

8. 2 m Relative Projection Height • Relative projection height = 2 m

9. 3 m 1.5 m Relative Projection Height • Relative projection height = -1.5 m

10. Optimum Angle of Projection (assuming there is no air resistance) • If Relative Projection Height = 0, the optimum angle = 450 • If Relative Projection Height > 0 , the optimum angle < 450 • If Relative Projection Height < 0 , the optimum angle > 450

11. Optimum Angle of Projection Relative Projection Height = 0 2 m Relative Projection Height = 2 m Text page 340 3 m 1.5 m Relative Projection Height = - 1.5 m

12. The Components of Speed Of Projection The velocity at any instant in the trajectory of a projectile can be represented as a vector that is tangent to the trajectory.

13. The Components of Speed Of Projection By finding the vertical and horizontal components for the instantaneous velocity vectors, you can find the instantaneous vertical and horizontal velocities.

14. The Components of Speed Of Projection EXAMPLE: A ball is thrown upward with a speed of projection of 20 m/s. If the angle of projection is 400, calculate the horizontal and vertical components of the speed of projection. S = Speed of projection = 20 m/s SH = Horizontal component SH = (S)(cos 400) = (20 m/s)(cos 400) SH = 15.32 m/s SV = Vertical component SV = (S)(sin 400) = (20 m/s)(sin 400) SV = 12.86 m/s SV 400 SH

15. Perpendicular Vectors Don’t Directly Affect Each Other • For example, if the projection angle of a projectile is horizontal (the vertical component of the projection speed is 0), it will fall as quickly as if it is dropped with a projection speed of 0.

16. Perpendicular Vectors Don’t Directly Affect Each Other Because the pull of gravity is unaffected by horizontal velocity, a projectile thrown horizontally has the same vertical velocity as an object dropped straight down. If the objects are released from the same height they will hit the ground at the same time (neglecting the effects of air resistance).

17. Acceleration Due to Gravity in Projectile Motion g (or ag) has the value of –9.81 m/s2 (metric units) or –32 ft/s2 (English units) when used in projectile motion calculations. Because the horizontal and vertical components of a trajectory don’t affect each other, if air resistance is neglected horizontal acceleration = 0 and vertical acceleration = g (or ag).

18. If Relative Projection Height = 0, the final angle and velocity of a projectile are equal in magnitude and opposite in direction to those of the projectile when it is released or launched (if air resistance is neglected).

19. q q If air resistance is neglected, the initial angle and final angle of the trajectory are the same.

20. Equations of Constant Acceleration • Formulas applied when acceleration is unchanging (as in the case of the acceleration due to gravity) Text p. 345 • v2 = v1 + at [This is derived from the basic formula: a = v/t = (v2 – v1)/t ] • d = v1t + (1/2)at2 • v22 = v12 + 2ad These formulas assume that: d= displacement, v1 = initial velocity, v2 = final velocity, a = acceleration, and t = time

21. PROJECTILE RANGE Assuming an object is released and lands at the same height and there is no air resistance: • VV2 = Vertical velocity at peak • t = Time to reach peak • ttotal = Total flight time = 2t • = Angle of projection aV = Vertical Acceleration = -ag V = Initial projectile velocity VV1 = Initial vertical velocity VH = Horizontal Velocity VV2 = 0 V aV t VV1  VH Range

22. PROJECTILE RANGE VH = Vcos  VV1 = Vsin  aV = -ag = (VV2 – VV1)/t t = (VV2 – VV1)/ -ag = (0 – VV1)/ -ag = VV1/ ag = Vsin  / ag ttotal = 2t = 2(Vsin  / ag) Range = (VH)(ttotal) = 2(Vcos )(Vsin )/ag) = 2(V2cos sin )/ag VV2 = 0 V aV t VV1  VH Range

23. Air Resistance and Projectile Motion Air resistance (or air drag) will tend to affect the velocity of a projectile. It tends to slow down the horizontal component of velocity so that the path of a projectile (if the initial horizontal component  0) will tend to have a steeper angle at the end than when the projectile is launched.

24. Without Air Resistance With Air Resistance Air Resistance and Projectile Motion Air resistance will tend to cause a projectile to fall shorter than it would if there were no air resistance.