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• Monkey and hunter experiment

4.1 Independence of horizontal and vertical motions. • Body projected horizontally under gravity. • Monkey and hunter experiment. • Body projected at an angle q under gravity. Kicking a football. 4.1 Independence of horizontal and vertical motions (SB p. 144).

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• Monkey and hunter experiment

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  1. 4.1 Independence of horizontal and vertical motions • Body projected horizontally under gravity •Monkey and hunter experiment • Body projected at an angle q under gravity

  2. Kicking a football 4.1 Independence of horizontal and vertical motions (SB p. 144) What is “projectile motion”? The football will be projected along____________ path

  3. 4.1 Independence of horizontal and vertical motions (SB p. 144) Independence of horizontal andvertical motions • Assume air resistance is negligible. During the flight, • the horizontal velocity ______________ • _______________ • (2) the vertical velocity is subjected to a _________________________

  4. breaks the aluminium strip and the circuit of the electromagnet bullet leaves the mouth of the toy gun iron monkey falls vertically downwards 1 2 3 4 the bullet always hits the monkey 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment

  5. The bullet and the monkey start with _______ vertical downward velocity They fall under the same acceleration due to gravity their vertical motions are __________ Their _______ positions are the same at any instant of time 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment Why does the bullet always hit the monkey?

  6. The horizontal and vertical components of a projectile motion are ____________ of each other. 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment We can conclude that:

  7. a marble is projected horizontally off the edge of a table travels with a uniform velocity in the horizontal direction and accelerates uniformly downwards due to gravity the horizontal velocity remains constant 4.1 Independence of horizontal and vertical motions (SB p. 146) Body projected horizontally under gravity

  8. horizontal motion vertical motion They are independent of each other 4.1 Independence of horizontal and vertical motions (SB p. 146) Body projected horizontally under gravity

  9. Initial vertical velocity: uy = u sinq the object is projected with a velocity u at an angle q to the horizontal Initialhorizontal velocity: ux = _________ 4.1 Independence of horizontal and vertical motions (SB p. 152) Body projected at an angle qunder gravity

  10. 4.1 Independence of horizontal and vertical motions (SB p. 152) Body projected at an angle qunder gravity • The vertical motion of the object undergoes constant acceleration a = –g • Vertical component of velocity (vy) and the vertical displacement (y) at time t :

  11. 4.1 Independence of horizontal and vertical motions (SB p. 152) Body projected at an angle qunder gravity • The horizontal motion is uniform and the object moves with constant velocity • Horizontal velocity component (vx) and the horizontal displacement (x) at time t:

  12. 4.1 Independence of horizontal and vertical motions (SB p. 152) Body projected at an angle qunder gravity • Instantaneous velocity (v): • The direction of motion at certain instant:

  13. 4.1 Independence of horizontal and vertical motions (SB p. 153) Body projected at an angle qunder gravity At B, the object reaches its maximum height H with vy = ___,

  14. 4.1 Independence of horizontal and vertical motions (SB p. 153) Body projected at an angle qunder gravity At D, the time of flight of the object (T)

  15. 4.1 Independence of horizontal and vertical motions (SB p. 153) Body projected at an angle qunder gravity Horizontal displacement OD of the projectile is the range R

  16. The equation is of the parabolic form y = ax – bx2 trajectory of the projectile is a __________ 4.1 Independence of horizontal and vertical motions (SB p. 153) Body projected at an angle qunder gravity By combining: where 0o < < 90o

  17. occurs when sin2 = _______  = ____ 4.1 Independence of horizontal and vertical motions (SB p. 154) Body projected at an angle qunder gravity Note: 1.The vertical and horizontal motions of a projectile motion are independent of each other 2. The trajectory of the projectile is ___________ 3. The maximum value of R is _______  maximum range is obtained if the object is projected at an angle ____ to the horizontal

  18. two angles of projection for a given range R with a specific speed, except for  = 45o Maximum range at  = 45o Range at 60o = Range at 30o 4.1 Independence of horizontal and vertical motions (SB p. 154) Body projected at an angle qunder gravity Note: 4.  sin2 = sin(180o – 2)

  19. A projectile is launched with an initial velocity of 30 m s-1 at an angle of 60o above the horizontal. Calculate the magnitude and direction of its velocity 5 s after launch. (Take g = 9.8 m s-2) Ans : 27.5 m s-1, 56.9o below the horizontal 

  20. A police officer is chasing you across a roof top, both are running at 4.5 m s-1. Before you reaches the edge of the roof, you has to decide whether or not to try jumping to the roof of the next building, which is 6.2 m away but 4.8 m lower. Can you make it?

  21. Effect of Atmosphere In an projectile experiment, the launch angle is 60 and the launch speed is 100 mi/h and the following results are obtained : Path I (Vacuum)Path II (Air) Range 581 ft 323 ft Maximum height 252 ft 174 ft Time of flight 7.9 s 6.6 s Effects of Air Resistance : (i) (ii) (iii) (iv) The trajectory becomes asymmetric.

  22. 4.1 Independence of horizontal and vertical motions (SB p. 154) Body projected at an angle qunder gravity Note: 5. In real situation, air resistance reduces the speed and the range of the projectile Go to More to Know 1 Go to Example 3

  23. End

  24. 4.1 Independence of horizontal and vertical motions (SB p. 148) Q: An aeroplane, flying in a straight line at a constant height of 500 m with a speed of 200 m s–1, drops an object. The object takes a time t to reach the ground and travels a horizontal distance d in doing so. Taking g as 10 m s–2 and ignoring air resistance, what are the values of t and d ? Example 1 Solution

  25. When the object is released from the aeroplane,Horizontal component of velocity = Velocity of the aeroplane= 200 m s–1For the vertical motion of the object:Initial velocity = 0, acceleration = g = 10 m s–2, displacement = 500 m Using 500 = 0 +  10  t2 t2 = 100t = 10 s For the horizontal motion:Distance (d) = 200 t = 200  10 = 2 000 m 4.1 Independence of horizontal and vertical motions (SB p. 148) Solution: Example 1 Return to Text

  26. 4.1 Independence of horizontal and vertical motions (SB p. 149) Q: When a rifle is fired horizontally at a target P on a screen at a range of 25 m, the bullet strikes the screen at a point 5.0 mm below P. The screen is now moved to a distance of 50 m from the rifle and the rifle again is fired horizontally at P in its new position.Assuming that air resistancemay be neglected, what is the new distance below P at which the screen would now be struck? Example 2 Solution

  27. 4.1 Independence of horizontal and vertical motions (SB p. 149) Solution:Let v = velocity of the bullet when it leaves the rifle. Time taken for the bullet to travel through a horizontal distance of 25 m (t1) = Consider vertical motion of the bullet,initial velocity (u) = 0, acceleration (a) = g, displacement (s) = 5.0  10–3 m, time (t1) =Using Example 2

  28. 4.1 Independence of horizontal and vertical motions (SB p. 149) Solution (cont’d):With the screen at 50 m from the rifle, time of travel (t2) =UsingNew distance below P (h) = (2)  (1) Example 2 Return to Text

  29. More to Know 1 So far, air resistance has been neglected for projectile motions. In real situations, the trajectory of the projectile is influenced by air resistance in the extent that depends on the mass, shape and size of the objects. 4.1 Independence of horizontal and vertical motions (SB p. 154) Return to Text

  30. 4.1 Independence of horizontal and vertical motions (SB p. 154) Q: An aeroplane flies at a height h with a constant horizontal velocity u so as to fly over a cannon. When the aeroplane is directly over the cannon, a shell was fired to hit the aeroplane. Neglecting air resistance, what is the minimum speed of the shell in order to hit the aeroplane? (g = acceleration due to gravity) Example 3 Solution

  31. 4.1 Independence of horizontal and vertical motions (SB p. 154) Solution:Let v = minimum speed of shell, = angle of projection.The minimum speed is when the shell will hit the aeroplane at the maximum height of the shell trajectory. Since the shell is fired when the aeroplane is directly above it, the horizontal component of velocity of the shell must be the same as the velocity of the aeroplane, i.e. same u. Example 3

  32. 4.1 Independence of horizontal and vertical motions (SB p. 154) Solution (cont’d):Using the equation,Horizontal component of velocity of the shell, Example 3 Return to Text

  33. 4.2 Terminal velocity (Vertical motion under gravity with air resistance)

  34. falling under influence of air resistance net force acting on him is due to his weight (mg) and the upward air resistance (f) By applying Newton’s Second Law: ma = mg – f 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air

  35. the air resistance acting on him becomes larger where b = constant f = bv 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air • his velocity (v) increases

  36. = his weight net force acting on him = 0 reaches the terminal velocity (vTA) with 0 acceleration the air resistance 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air • moves faster and faster

  37. reaches terminal velocity (vTA) with 0 acceleration 0 = mg - bvTA 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air

  38. air resistance > parachutist’s weight resultant upward net force acting on him slows him down air resistance 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air

  39. the air resistance will be balanced by the weight again at a new lower terminal velocity vTC before he lands safely 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air

  40. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 158) A parachutist falling in the air

  41. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions 1. Assume air resistance is negligible, in a projectile motion, the horizontal velocity remains constant, and the vertical velocity is subjected to a constant acceleration due to gravity. 2. The horizontal velocity (vx) and vertical velocity (vy) are independent of each other in projectile motions.

  42. 3. For objects projected horizontally under gravity:(a) The instantaneous velocity (v) of the projectile is: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions

  43. (b) The direction of motion is determined by: (c) If the projectile is projected at a height H above the ground, its time of flight (tf) is: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions

  44. (d) The total horizontal displacement travelled by the object is called the range (R): where ux is the initial horizontal velocity (e) The trajectory of the projectile is a parabola: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions

  45. 4. For objects projected with initial velocity (u) at an angle  under gravity:(a) Its maximum height (H) is: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions

  46. (b) Its time of flight (tf) is: (c) Its range (R) is: (d) The trajectory of the projectile is a parabola: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.1 Independence of horizontal and vertical motions

  47. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.2 Terminal velocity 5. Consider a parachutist of mass m falling under the influence of air resistance. As the parachutist falls, its velocity (v) increases. When the velocity increases, the air resistance acted on the parachutist also increases. We have: f = bv where f is the air resistance and b is a constant.

  48. 6. The terminal velocity of the parachutist (vT) is: where g is the acceleration due to gravity. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) Summary 4.2 Terminal velocity

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