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

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

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3 4.1 Independence of horizontal and vertical motions (SB p. 144) Independence of horizontal and vertical motions Assume air resistance is negligible. During the flight, (1)the horizontal velocity ______________ _______________ (2)the vertical velocity is subjected to a _________________________

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5 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment bullet leaves the mouth of the toy gun1 breaks the aluminium strip and the circuit of the electromagnet 2 iron monkey falls vertically downwards 3 the bullet always hits the monkey 4

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6 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment 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 Why does the bullet always hit the monkey?

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7 4.1 Independence of horizontal and vertical motions (SB p. 145) Monkey and hunter experiment The horizontal and vertical components of a projectile motion are ____________ of each other. We can conclude that:

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8 4.1 Independence of horizontal and vertical motions (SB p. 146) Body projected horizontally under gravity 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

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9 4.1 Independence of horizontal and vertical motions (SB p. 146) Body projected horizontally under gravity horizontal motion vertical motion They are independent of each other

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11 4.1 Independence of horizontal and vertical motions (SB p. 152) Body projected at an angle under gravity Initial vertical velocity: u y = u sin Initial horizontal velocity: u x = _________ the object is projected with a velocity u at an angle to the horizontal

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12 4.1 Independence of horizontal and vertical motions (SB p. 152) The vertical motion of the object undergoes constant acceleration a = –g Vertical component of velocity (v y ) and the vertical displacement (y) at time t : Body projected at an angle under gravity

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13 4.1 Independence of horizontal and vertical motions (SB p. 152) The horizontal motion is uniform and the object moves with constant velocity Horizontal velocity component (v x ) and the horizontal displacement (x) at time t: Body projected at an angle under gravity

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

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15 4.1 Independence of horizontal and vertical motions (SB p. 153) At B, the object reaches its maximum height H with v y = ___, Body projected at an angle under gravity

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16 4.1 Independence of horizontal and vertical motions (SB p. 153) At D, the time of flight of the object (T) Body projected at an angle under gravity

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17 4.1 Independence of horizontal and vertical motions (SB p. 153) Horizontal displacement OD of the projectile is the range R Body projected at an angle under gravity

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18 By combining: 4.1 Independence of horizontal and vertical motions (SB p. 153) The equation is of the parabolic form y = ax – bx 2 trajectory of the projectile is a __________ where 0 o < < 90 o Body projected at an angle under gravity

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19 4.1 Independence of horizontal and vertical motions (SB p. 154) occurs when sin2 = _______ = ____ maximum range is obtained if the object is projected at an angle ____ to the horizontal 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 _______ Body projected at an angle under gravity

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20 4.1 Independence of horizontal and vertical motions (SB p. 154) Note: 4. sin2 = sin(180 o – 2 ) Maximum range at = 45 o Range at 60 o = Range at 30 o two angles of projection for a given range R with a specific speed, except for = 45 o Body projected at an angle under gravity

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21 A projectile is launched with an initial velocity of 30 m s -1 at an angle of 60 o 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, 56.9 o below the horizontal

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22 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?

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23 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) Range581 ft323 ft Maximum height252 ft174 ft Time of flight7.9 s6.6 s Effects of Air Resistance : (i) (ii) (iii) (iv)The trajectory becomes asymmetric.

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24 4.1 Independence of horizontal and vertical motions (SB p. 154) Note: 5. In real situation, air resistance reduces the speed and the range of the projectile Go to More to Know 1 More to Know 1 Go to Example 3 Example 3 Body projected at an angle under gravity

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26 4.1 Independence of horizontal and vertical motions (SB p. 148) Q: 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 ? Solution

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27 4.1 Independence of horizontal and vertical motions (SB p. 148) Solution: For the horizontal motion: Distance (d) = 200 t = 200 10 = 2 000 m When the object is released from the aeroplane, Horizontal component of velocity = Velocity of the aeroplane = 200 m s –1 For the vertical motion of the object: Initial velocity = 0, acceleration = g = 10 m s –2, displacement = 500 m Using 500 = 0 + 10 t 2 t 2 = 100 t = 10 s Return to Text

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28 Q: 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 resistance may be neglected, what is the new distance below P at which the screen would now be struck? Solution 4.1 Independence of horizontal and vertical motions (SB p. 149)

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29 4.1 Independence of horizontal and vertical motions (SB p. 149) Solution: 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 (t 1 ) = Consider vertical motion of the bullet, initial velocity (u) = 0, acceleration (a) = g, displacement (s) = 5.0 10 –3 m, time (t 1 ) = Using

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30 4.1 Independence of horizontal and vertical motions (SB p. 149) Solution (cont’d): Solution (cont’d): With the screen at 50 m from the rifle, time of travel (t 2 ) = Using New distance below P (h) = (2) (1) Return to Text

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31 4.1 Independence of horizontal and vertical motions (SB p. 154) 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. Return to Text

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32 Q: 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) 4.1 Independence of horizontal and vertical motions (SB p. 154) Solution

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33 Solution: Solution: Letv = 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. 4.1 Independence of horizontal and vertical motions (SB p. 154)

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

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35 4.2 Terminal velocity (Vertical motion under gravity with air resistance)

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36 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

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37 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)

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38 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 = 0 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 157)

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39 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)

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40 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)

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41 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)

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

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43 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)

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44 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)

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45 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)

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46 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)

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47 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)

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48 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)

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49 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)

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50 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)

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51 4.2 Terminal velocity (Vertical motion under gravity with air resistance (SB p. 160)

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CHAPTER 6 MOTION IN 2 DIMENSIONS.

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