PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting...

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PROJECTILE MOTION

Transcript of PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting...

Page 1: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

PROJECTILE MOTION

Page 2: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

When we throw a ball :

•  There is a constant velocity horizontal motion

•  And there is an accelerated vertical motion

•  These components act independently of each other

Page 3: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

PROJECTILE MOTION

•  A falling object with constant linear velocity and vertical acceleration :

Page 4: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

2-D motion •  The path or trajectory projectiles make is

parabolic (neglecting air resistance). •  Two independent motions- horizontal and

vertical. •  Use kinematics equations in one direction

at a time. •  The connection between the two motions is

the variable time.

Page 5: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Projectile motion • Vertical motion is free fall-

constant acceleration motion. becomes…

Δx = vi t + ( ½) a t 2

vf = vi + at

vf2 = vi

2 + 2aΔx

Page 6: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Projectile motion • Horizontal motion is constant

velocity motion.

..and ax= 0, so Δx = vi t + ( ½) a t2

becomes

Page 7: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Problem Type 1:

•  A projectile is launched with an initial horizontal velocity from an elevated position and follows a parabolic path to the ground. Predictable unknowns include the initial speed of the projectile, the initial height of the projectile, the time of flight, and the horizontal distance of the projectile.

Page 8: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Problem Type 2:

•  A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. Upon reaching the peak, the projectile falls with a motion which is symmetrical to its path upwards to the peak. Predictable unknowns include the time of flight, the horizontal range, and the height of the projectile when it is at its peak.

Page 9: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Type 1 examples

•  A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location.

Page 10: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

List Givens

•  Horizontal (x) info:

•  x = ? •  vix = 2.4 m/s •  ax = 0 m/s2

Equations:

•  Vertical (y) info:

•  y = -0.60 m •  viy = 0 m/s •  ay = -9.8 m/s2

Δx = vi t + ( ½) a t2 vf = vi + at vf

2 = vi2 + 2aΔx

A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and

the ball's landing location.

t = ?

Page 11: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Choose an equation and solve

•  y = viy t + ( ½) ay t2 •  -0.60 m = (0 m/s)•t + 0.5•(-9.8 m/s/s)•t2

•  -0.60 m = (-4.9 m/s/s)•t2

•  0.122 s2 = t2

•  t = 0.350 s

Page 12: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Now use equation in horizontal direction to solve for x

•  x = vixt + 0.5axt2 •  x = (2.4 m/s)(0.3499 s) + 0.5•(0)•(0.3499 s)2

•  x = (2.4 m/s)•(0.3499 s) •  x = 0.84 m

Page 13: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Example B, problem type 1

•  A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

Page 14: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

List Givens •  Horizontal (x) info:

•  x = 35 m •  vix = ? •  ax = 0 m/s2

•  Vertical (y) info:

•  y = -22.0 m •  viy = 0 m/s •  ay = -9.8 m/s2

A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

Δx = vi t + ( ½) a t2 vf = vi + at vf

2 = vi2 + 2aΔx

Page 15: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Solve

•  y = viyt + 0.5ayt2 •  t= 2.12 s

•  Now use x = vixt + 0.5axt2 to solve for vix •  vix = 16.5 m/s

Page 16: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Previous Examples = Problem Type 1

Page 17: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Problem Type 2

Page 18: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Projectiles launched at an angle

•  initial velocity = vi •  launch angle θ •  separate initial velocity into

components

θ

vi vyi

vx

Page 19: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Components from trig. func.

• Use sine and cosine functions to find components.

Page 20: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Problem type 2 (example A)

•  A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the football.

Page 21: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

back to the problem…

A football is kicked with an initial velocity of 25 m/s at an angle of 45o with the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the football.

•  HORIZONTAL COMPONENT

•  vix = vi • cosθ •  vix = 25 m/s • cos 45o •  vix = 17.7 m/s

•  VERTICAL COMPONENT

•  viy = vi • sinθ •  viy = 25 m/s • sin 45o •  viy = 17.7 m/s

Page 22: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

List Givens

•  Horizontal (x) info:

•  x = ??? •  vix = 17.7 m/s •  vfx = 17.7 m/s •  ax = 0 m/s2

Equations:

•  Vertical (y) info:

•  y = ??? •  viy = 17.7 m/s •  vfy = -17.7 m/s ** •  ay = -9.8 m/s2

A football is kicked with an initial velocity of 25 m/s at an angle of 45o with the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the football.

Δx = vi t + ( ½) a t2 vf = vi + at vf

2 = vi2 + 2aΔx

Page 23: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

*symmetry of projectile motion

Page 24: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Same concept of symmetry applies to free fall motion

Page 25: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Use vertical info to find t

vfy = viy + ayt

-17.7 m/s = 17.7 m/s + (-9.8 m/s2) t

-35.4 m/s = (-9.8 m/s2) t

3.6077 s = t = 3.61 s

Page 26: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Now find horizontal distance

•  x = vix t + ½ axt2

•  x = (17.7 m/s)(3.6077 s) + ½ (0 m/s/s)(3.6077 s)2

•  x = (17.7 m/s)•(3.6077 s)

•  x = 63.8 m

Page 27: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Find “height of projectile at its peak” (in other words, the vertical displacement, y)

•  Note: The height is reached when at a time= ½ t

•  y = viyt + ½ ayt2

•  y = (17.7 m/s)(1.80 s) + ½ (-9.8 m/s/s)(1.80 s)2

•  y = 31.9 m + (-15.9 m)

•  y = 15.9 m

Page 28: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Find “height of projectile at its peak” (in other words, the vertical displacement, y)

•  Note: You could also use vf2 = vi

2 + 2aΔy as long as you know vfy = 0 at the peak height

Page 29: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Type 2: example B

•  A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28o above the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the long-jumper.

Page 30: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28o above the horizontal. Determine the time of flight, the

horizontal distance, and the peak height of the long-jumper.

•  HORIZONTAL COMPONENT

•  vix = vi • cosθ •  vix = 12 m/s • cos 28o •  vix = 10.6 m/s

•  VERTICAL COMPONENT

•  viy = vi • sinθ •  viy = 12 m/s • sin 28o •  viy = 5.6 m/s

Page 31: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

List Givens

•  Horizontal (x) info:

•  x = ??? •  vix = 10.6 m/s •  vfx = 10.6 m/s •  ax = 0 m/s2

Equations:

•  Vertical (y) info:

•  ypeak = ??? •  viy = 5.6 m/s •  vfy = -5.6 m/s •  vypeak = 0 m/s •  ay = -9.8 m/s2

A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the horizontal. Determine the time of flight,

the horizontal distance, and the peak height of the long-jumper.

Δx = vi t + ( ½) a t2 vf = vi + at vf

2 = vi2 + 2aΔx

Page 32: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Use vertical info to find t

•  vfy = viy + ayt

•  -5.6 m/s = 5.6 m/s + (-9.8 m/s2) t

•  -11.2 m/s = (-9.8 m/s2) t

•  1.1497 s = t = 1.1 s

Page 33: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Now find horizontal distance

•  x = vix t + ½ axt2

•  x = (10.6 m/s)(1.1497 s) + ½ (0 m/s/s)(1.1497 s)2

•  x = (10.6 m/s)•(1.1497 s)

•  x = 12.2 m

Page 34: PROJECTILE MOTION A.P..pdf · • The path or trajectory projectiles make is parabolic (neglecting air resistance). • Two independent motions- horizontal and vertical. • Use kinematics

Find “height of projectile at its peak” (in other words, the vertical displacement, y)

•  Note: The height is reached when at a time= ½ t

•  y = viyt + ½ ayt2

•  y = (5.6 m/s)(0.5748 s) + ½ (-9.8 m/s2)(0.5748 s)2

•  y = 3.219 m + (-1.6189 m)

•  y = 1.6 m