Since a projectile moves in 2-dimensions, it therefore has 2 components just like a resultant vector: Horizontal Vertical
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3 Since a projectile moves in 2-dimensions, it therefore has 2 components just like a resultant vector: Horizontal Vertical
4 With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest.
5 Velocity The velocity of a projectile is shown at various points along its path. Notice that the vertical component changes while the horizontal component does not. Air resistance is neglected.
6 Height For the component vectors of the angled projectile motion, the horizontal component is always the same and only the vertical component changes.
7 Height At the top of the path the vertical component shrinks to zero.
8 Together, these components produce what is called a trajectory or path. This path is parabolic in nature. Component Magnitude Direction Horizontal Constant Constant Vertical Changes Changes
9 Range The angle at which the projectile is launched affects the distance that it travels.
10 Horizontal Ranges Projectiles that are launched at the same speed but at different angles reach different heights (altitude) above the ground. They also travel different horizontal distances, that is, they have different horizontal ranges.
11 Horizontal Range Both projectiles have the same launching speed. The initial velocity vector has a greater vertical component than when the projection angle is less. This greater component results in a higher path. The horizontal component is less, so the range is less.
12 The paths of projectiles launched at the same speed but at different angles. The paths neglect air resistance.
13 The same range is obtained for two different projection angles angles that add up to 90. An object thrown into the air at an angle of 60 will have the same range as at 30 with the same speed.
14 Max Range Maximum range is attained at an angle of 45
15 Max Height Without air resistance, a projectile will reach maximum height in the same time it takes to fall from that height to the ground.
16 Acceleration The deceleration due to gravity going up is the same as the acceleration due to gravity coming down.
17 Speed The projectile hits the ground with the same speed it had when it was projected upward from the ground.
18 Thinker! A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration? Answer Vertical acceleration = m/s 2 or 32 ft/s 2 Horizontal acceleration = zero
19 Thinker! At what point in its path does a projectile have minimum speed? Answer The minimum speed of a projectile occurs at the top of its path. If it is launched vertically, its speed at the top is zero. If it is projected at an angle, the vertical component of velocity is still zero at the top, leaving only the horizontal component.
20 NO V y at the top of the trajectory. V y decreases on the way upward V y increases on the way down, V x is constant Component Magnitude Direction Horizontal Constant Constant Vertical Decreases up, top, Increases down Changes
21 Projectiles launched at angles summary: The horizontal velocity is constant. It rises and falls in equal time intervals. It reaches maximum height in half the total time. Gravity only affects the vertical motion.
22 Components Since the projectile was launched at an angle, the velocity MUST be broken into components!!! v iy v i θ v ix V iy = V i sin θ V ix = V i cos θ
23 There are several things you must consider when doing these types of projectiles besides using components. If it begins and ends at ground level, the d y displacement is ZERO: d y = 0
24 Formulas You will still use the kinematic equations, but YOU MUST use COMPONENTS in the equation. v iy v i θ v ix
25 What is the first thing you need to do if a projectile is launched at an angle? You must break the initial velocity into horizontal and vertical components
26 Maximum Height d ymax Maximum Range d xmax = range
27 Question #1 A volleyball hit into the air has an initial speed of 10 meters per second. Which vector best represents the angle above the horizontal that the ball should be hit to remain in the air for the greatest amount of time?
28 Question #2 A vector makes an angle, θ, with the horizontal. The horizontal and vertical components of the vector will be equal in magnitude if angle θ is: a. 30 b. 45 c. 60 d. 90
29 Question #2 A vector makes an angle, θ, with the horizontal. The horizontal and vertical components of the vector will be equal in magnitude if angle θ is: a. 30 b. 45 c. 60 d. 90
30 Question #3 A place kicker kicks a football with a velocity of 20 m/s and at an angle of 53. a. What are the horizontal and vertical components of the initial velocity? b. How long is the ball in the air? c. How far away does it land? d. How high does it travel?
31 Example #4 A body is projected upward from the level ground at an angle of 50 with the horizontal has an initial speed of 40 m/s. a. What are the horizontal and vertical components of the initial velocity? b. How long will it be before it hits the ground? c. How far from the starting point will the object hit the ground? d. What is the maximum height it reached in the air? a. V ix = m/s; V iy = m/s b s c. d x = m d. d ymax = m
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