Chapter 3 2-D Motion

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1 Chapter 3 2-D Motion

2 We will need to use vectors and their properties a lot for this chapter.

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14 Pythagorean Theorem:

15 Sample problem: First you hike 100 m north. Then hike 50 m west. Finally hike 100 meters 45 degrees northeast. Find your final total displacement using algebra and by geometry (drawing pictures).

16 First hike 100 m north

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22 Direction is about 26.5 degrees below the negative x-axis. Or about degrees from the positive x-axis.

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26 V i V f a t

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29 You re driving a boat 4 m/s with a heading of 45

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32 Projectile Motion Objects that move in both the x and y directions. Examples: thrown and batted baseballs bullets and arrows cannonballs water from water fountains animals jumping See figure 3.14

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34 We will observe objects propelled: sideways directly upwards at an angle simply dropped We will see, that no matter the initial conditions (the initial velocity vector), the laws and formulas we use, apply to all cases.

35 Example: Horizontal Gunshot If you shoot a gun horizontally. The bullet will leave the gun with some velocity. The horizontal component will stay constant, (the initial velocity). The vertical component of velocity will begin to change (increase downwards) due to gravity. This is why the bullet hits the target a bit lower than where the gun was aimed. See figures 3.16, 3.17

36 Projectile motion simulator Projectiles/Projectile-Simulator/Projectile-Simulator-Interactive Play around with these to see the effects of changing some of the variables. Might be useful for homework.

37 Effect of varying the initial launch angle V o is constant Notice what happens for complementary angles.

38 Two cannonballs shot at the same time, with the same speed, but at two different angles. Which cannonball hits the target first?

39 Two cannonballs shot at the same time, with the same speed, but at two different angles. Which cannonball hits the target first? The cannonball that follows the bottom path hits first. It will need a higher x-component of velocity to hit the target.

40 Applying equations of motion to 2-D Motion The equations of motion for constant acceleration from chapter 2 are valid separately for both, motion in the x and y directions. However the initial velocity now has 2 components (x and y). If a projectile has an initial velocity at an angle θ 0 with respect to the horizontal axis, θ 0 is known as the projection angle. The components of the initial velocity are: v 0x = v 0 cos θ 0 v 0y = v 0 sin θ 0

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43 Refer again to figure 3.14

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45 Using the range equation, can you determine which projection angle, leads to the biggest range, given a constant initial velocity?

46 For projectile motion: Reminder: Horizontal motion and vertical motion are independent. Horizontal motion is constant. Vertical motion is accelerated. Unless the horizontal motion has some sort of constraint, it will be the motion in the vertical direction that determines how long a projectile is in the air.

47 The time that a projectile is in the air depends on the vertical displacement and the vertical component of velocity. Here we roll a ball off a table and drop a ball at the same time.

48 Examples A rescue plane drops a package while the plane is travelling horizontally at 40 m/s, and an elevation of 100 meters. What is the horizontal distance the package travels while falling? What are the horizontal and vertical components of the velocity when it lands?

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50 Find the horizontal and vertical components of the velocity when the package lands. Horizontal component is constant v fx = v ox = 40 m/s Now want vertical component of velocity: use: v y = v 0y g t v y = 0 (9.8 m/s 2 )(4.5 s) = m/s Could also use v y2 = v 0y2 2gy v y2 = -2(9.8 m/s 2 )(-100 m) v y = 44.3 m/s This method didn t give the minus sign for direction. Why are the two results slightly off???

51 Cannonball Example A cannon can shoot a cannonball with a velocity of 150 m/s. If the cannonball is shot at an angle of 30 degrees, how far will it travel? (Assuming it lands at the same level it is shot.)

52 Cannonball Example, alternative method

53 How much farther will the cannon shoot, if the projection angle is 45 degrees?

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59 Rocket launched from a plane Jet plane traveling horizontally at 100 m/s drops a rocket from a high altitude. The rocket immediately fires its engines and accelerates 20 m/s 2 in the x-direction, while it falls due to gravity in the y-direction. After the rocket has fallen 1 km, find: a) the y-velocity b) x-velocity c) magnitude and direction of the velocity

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62 The negative sign means down.

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65 A baseball pitched at 101 mph is thrown horizontally. How far would the ball fall vertically by the time it reaches home plate, 60.5 feet away m

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