3.3 Projectile Motion

1. 3.3 Projectile Motion • The motion of an object under the influence of gravity only • The form of two-dimensional motion

2. Assumptions of Projectile Motion • The free-fall acceleration is constant over the range of motion • And is directed downward • The effect of air friction is negligible • With these assumptions, the motion of the object will follow

3. Projectile Motion Vectors • The final position is the vector sum of the initial position, the displacement resulting from the initial velocity and that resulting from the acceleration • This path of the object is called the trajectory Fig 3.6

4. Analyzing Projectile Motion • Consider the motion as the superposition of the motions in the x- and y-directions • Constant-velocity motion in the x direction • ax = 0 • A free-fall motion in the y direction • ay = -g

5. Verifying the Parabolic Trajectory • Reference frame chosen • y is vertical with upward positive • Acceleration components • ay = -g and ax = 0 • Initial velocity components • vxi = vi cos qi and vyi = vi sin qi

6. Projectile Motion – Velocity at any instant • The velocity components for the projectile at any time t are: • vxf = vxi = vi cos qi = constant • vyf = vyi – g t = vi sin qi – g t

7. Projectile Motion – Position • Displacements • xf = vxit = (vi cos qi)t • yf = vyit + 1/2ay t2 = (vi sinqi)t - 1/2 gt2 • Combining the equations gives: • This is in the form of y = ax – bx2 which is the standard form of a parabola

8. What are the range and the maximum height of a projectile • The range, R, is the maximum horizontal distance of the projectile • The maximum height, h, is the vertical distance above the initial position that the projectile can reaches. Fig 3.7

9. Projectile Motion Diagram Fig 3.5

10. Projectile Motion – Implications • The y-component of the velocity is zero at the maximum height of the trajectory • The accleration stays the same throughout the trajectory

11. Height of a Projectile, equation • The maximum height of the projectile can be found in terms of the initial velocity vector: • The time to reach the maximum:

12. Range of a Projectile, equation • The range of a projectile can be expressed in terms of the initial velocity vector: • The time of flight = 2tm • This is valid only for symmetric trajectory

13. More About the Range of a Projectile

14. Range of a Projectile, final • The maximum range occurs at qi = 45o • Complementary angles will produce the same range • The maximum height will be different for the two angles • The times of the flight will be different for the two angles

15. Non-Symmetric Projectile Motion

21. Fig 3.10

30. 3.4 Uniform Circular Motion • Uniform circular motion occurs when an object moves in a circular path with a constant speed • An acceleration exists since the direction of the motion is changing • This change in velocity is related to an acceleration • The velocity vector is always tangent to the path of the object

31. Changing Velocity in Uniform Circular Motion • The change in the velocity vector is due to the change in direction • The vector diagram shows Fig 3.11

32. Centripetal Acceleration • The acceleration is always perpendicular to the path of the motion • The acceleration always points toward the center of the circle of motion • This acceleration is called the centripetal acceleration • Centripetal means center-seeking

33. Centripetal Acceleration, cont • The magnitude of the centripetal acceleration vector is given by • The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

34. Period • The period, T, is the time interval required for one complete revolution • The speed of the particle would be the circumference of the circle of motion divided by the period • Therefore, the period is

37. 3.5 Tangential Acceleration • The magnitude of the velocity could also be changing, as well as the direction • In this case, there would be a tangential acceleration

38. Total Acceleration • The tangential acceleration causes the change in the speed of the particle and is in the direction of velocity vector, which parallels to the line tangent to the path. • The radial acceleration comes from a change in the direction of the velocity vector and is perpendicular to the path. • At a given speed, the radial acceleration is large when the radius of curvature r is small and small when r is large.

39. Total Acceleration, equations • The tangential acceleration: • The radial acceleration: • The total acceleration:

40. 3.6 Relative Velocity • Two observers moving relative to each other generally do not agree on the outcome of an experiment • For example, the observer on the side of the road observes a different speed for the red car than does the observer in the blue car Fig 3.13

41. Relative Velocity, generalized • Reference frame S is stationary • Reference frame S’ is moving • Define time t = 0 as that time when the origins coincide Fig 3.14

42. Relative Velocity, equations • The positions as seen from the two reference frames are related through the velocity • The derivative of the position equation will give the velocity equation • This can also be expressed in terms of the observer O’

43. Fig 3.15(a)

47. Fig 3.15(b)

49. Exercises of chapter 3 • 2, 3,16, 20, 27, 30, 38, 46, 50, 54, 63

50. Chapter 31 Particle Physics