1© Manhattan Press (H.K.) Ltd. 4.2 Terminal velocity (Vertical motion under gravity with air resistance)

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1© Manhattan Press (H.K.) Ltd. 4.2 Terminal velocity (Vertical motion under gravity with air resistance)

2 © Manhattan Press (H.K.) Ltd. 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: ma = mg – f

3 © Manhattan Press (H.K.) Ltd. A parachutist falling in the air  his velocity (v) increases f = bv where b = constant the air resistance acting on him becomes larger 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157)

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

5 © Manhattan Press (H.K.) Ltd. A parachutist falling in the air 0 = mg - bv TA reaches terminal velocity (v TA ) with 0 acceleration 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157)

6 © Manhattan Press (H.K.) Ltd. A parachutist falling in the air air resistance > parachutist’s weightair resistance resultant upward net force acting on him slows him down 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157)

7 © Manhattan Press (H.K.) Ltd. A parachutist falling in the air at a new lower terminal velocity v TC before he lands safely the air resistance will be balanced by the weight again 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157)

8 © Manhattan Press (H.K.) Ltd. A parachutist falling in the air 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 158)

9 © Manhattan Press (H.K.) Ltd. 4.1Independence 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 (v x ) and vertical velocity (v y ) are independent of each other in projectile motions. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159)

10 © Manhattan Press (H.K.) Ltd. 4.1Independence of horizontal and vertical motions 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)

11 © Manhattan Press (H.K.) Ltd. 4.1Independence 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 (t f ) is: 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159)

12 © Manhattan Press (H.K.) Ltd. 4.1Independence of horizontal and vertical motions (d)The total horizontal displacement travelled by the object is called the range (R): where u x 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)

13 © Manhattan Press (H.K.) Ltd. 4.1Independence of horizontal and vertical motions 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)

14 © Manhattan Press (H.K.) Ltd. 4.1Independence of horizontal and vertical motions (b)Its time of flight (t f ) 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)

15 © Manhattan Press (H.K.) Ltd. 4.2Terminal 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. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159)

16 © Manhattan Press (H.K.) Ltd. 4.2Terminal velocity 6. The terminal velocity of the parachutist (v T ) is: where g is the acceleration due to gravity. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 159)

17 © Manhattan Press (H.K.) Ltd. 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 160)

18 © Manhattan Press (H.K.) Ltd. End