4 2 Terminal velocity Vertical motion under gravity

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 158)

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159)

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 160)

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4. 2 Terminal velocity (Vertical motion under gravity with air resistance) © Manhattan Press (H. K. ) Ltd. 1

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air 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: © Manhattan Press (H. K. ) Ltd. ma = mg – f 2

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air Q his velocity (v) increases the air resistance acting on him becomes larger f = bv © Manhattan Press (H. K. ) Ltd. where b = constant 3

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air • moves faster and faster the air resistance = his weight net force acting on him = 0 reaches the terminal velocity (v. TA) with 0 acceleration © Manhattan Press (H. K. ) Ltd. 4

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air reaches terminal velocity (v. TA) with 0 acceleration 0 = mg - bv. TA © Manhattan Press (H. K. ) Ltd. 5

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air resistance > parachutist’s weight resultant upward net force acting on him © Manhattan Press (H. K. ) Ltd. slows him down air resistance 6

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157) A parachutist falling in the air resistance will be balanced by the weight again at a new lower terminal velocity v. TC before he lands safely © Manhattan Press (H. K. ) Ltd. 7

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 158) A parachutist falling in the air © Manhattan Press (H. K. ) Ltd. 8

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 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. © Manhattan Press (H. K. ) Ltd. 9

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 1 Independence of horizontal and vertical motions 3. For objects projected horizontally under gravity: (a) The instantaneous velocity (v) of the projectile is: © Manhattan Press (H. K. ) Ltd. 10

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 1 Independence of horizontal and vertical motions (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: © Manhattan Press (H. K. ) Ltd. 11

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 1 Independence of horizontal and vertical motions (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: © Manhattan Press (H. K. ) Ltd. 12

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 1 Independence of horizontal and vertical motions 4. For objects projected with initial velocity (u) at an angle under gravity: (a) Its maximum height (H) is: © Manhattan Press (H. K. ) Ltd. 13

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 1 Independence of horizontal and vertical motions (b) Its time of flight (tf) is: (c) Its range (R) is: (d) The trajectory of the projectile is a parabola: © Manhattan Press (H. K. ) Ltd. 14

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 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. © Manhattan Press (H. K. ) Ltd. 15

4. 2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159) 4. 2 Terminal velocity 6. The terminal velocity of the parachutist (v. T) is: where g is the acceleration due to gravity. © Manhattan Press (H. K. ) Ltd. 16