motion in two dimensions chapter 7.2 projectile motion what is the path of a projectile as it moves...
TRANSCRIPT
![Page 1: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/1.jpg)
Motionin Two Dimensions
Chapter 7.2
![Page 2: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/2.jpg)
Projectile Motion
What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?
Yes, both are possible.
What forces act on projectiles? Only gravity, which acts only in the negative
y-direction. Air resistance is ignored in projectile motion.
![Page 3: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/3.jpg)
Choosing Coordinates & Strategy
For projectile motion: Choose the y-axis for vertical motion where
gravity is a factor. Choose the x-axis for horizontal motion. Since
there are no forces acting in this direction (of course we will neglect friction due to air resistance), the speed will be constant (a = 0).
Analyze motion along the y-axis separate from the x-axis.
If you solve for time in one direction, you automatically solve for time in the other direction.
![Page 4: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/4.jpg)
The Trajectory of a Projectile
•What does the free-body diagram look like for force?
Fg
![Page 5: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/5.jpg)
The Vectors of Projectile Motion
What vectors exist in projectile motion? Velocity in both the x and y directions. Acceleration in the y direction only.
vy (Increasing)
vx (constant)
ay
ax = 0
Why is the velocity constant in the x-direction?
•No force acting on it.
Why does the velocity increase in the y-direction?
•Gravity.
![Page 6: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/6.jpg)
Ex. 1: Launching a Projectile Horizontally
A cannonball is shot horizontally off a cliff with an initial velocity of 30 m/s. If the height of the cliff is 50 m: How far from the base of the cliff does
the cannonball hit the ground? With what speed does the cannonball
hit the ground?
![Page 7: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/7.jpg)
Diagram the problem
50mFg = Fnet a = -g
vi
vf = ?
vx
vyx = ?
![Page 8: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/8.jpg)
State the Known & Unknown
Known: xi = 0 vix = 30 m/s yi = 0 viy = 0 m/s a = -g y = -50 m
Unknown: x at y = -50 m vf = ?
![Page 9: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/9.jpg)
Perform Calculations (y)
y-direction: vy = -gt y = viyt – ½ gt2
Using the first formula above: vy = (-9.8 m/s2)(3.2 s) = 31 m/s
g
yt
2
ssm
mt 2.3
/81.9
5022
![Page 10: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/10.jpg)
Perform Calculations (x)
x-Direction x = vixt x = (30 m/s)(3.2 s) = 96 m from the
base. Using the Pythagorean Theorem:
v = vx2 + vy
2
v = (30 m/s)2 + (31 m/s)2 = 43 m/s
![Page 11: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/11.jpg)
Ex. 2: Projectile Motion above the Horizontal
A ball is thrown from the top of the Science Wing with a velocity of 15 m/s at an angle of 50 degrees above the horizontal.
What are the x and y components of the initial velocity? What is the ball’s maximum height? If the height of the Science wing is 12 m, where will the
ball land?
![Page 12: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/12.jpg)
Diagram the problem
y
x
vi = 15 m/s
= 50°
Fg = Fnet a = -g
Ground
12 m
x = ?
vi = 15 m/s
vix
viy
= 50°
![Page 13: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/13.jpg)
State the Known & Unknown
Known: xi = 0 yi = 12 m vi = 15 m/s = 50° a = -g
Unknown: ymax = ? t = ? x = ? viy = ? vix = ?
![Page 14: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/14.jpg)
Perform the Calculations (ymax)
y-direction: Initial velocity: viy = visin
viy = (15 m/s)(sin 50°)
viy = 11.5 m/s
Time when vfy = 0 m/s: vfy = viy – gt t = viy / g t = (11.5 m/s)/(9.81 m/s2) t = 1.17 s
Determine the maximum height: ymax = yi +viyt – ½ gt2
ymax = 12 m + (11.5 m/s)(1.17 s) – ½ (9.81 m/s2)(1.17 s)2
ymax = 18.7 m
vi = 15 m/s
vxi
vyi
= 50°
![Page 15: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/15.jpg)
Perform the Calculations (t)
Since the ball will accelerate due to gravity over the distance it is falling back to the ground, the time for this segment can be determined as follows
Time when ball hits the ground: ymax = viyt – ½ gt2
Since yi can be set to zero as can viy, t = 2*ymax/g t = 2(18.7 m)/ (9.81 m/s2) t = 1.95 s
By adding the time it takes the ball to reach its maximum height to the time it it takes to reach the ground will give you the total time.
ttotal = 1.17 s + 1.95 s = 3.12 s
![Page 16: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/16.jpg)
Perform the Calculations (x)
x-direction: Initial velocity: vix = vicos
vix = (15 m/s)(cos 50°)
vix = 9.64 m/s
Determine the total distance: x = vixt x = (9.64 m/s)(3.12 s) x = 30.1 m
vi = 15 m/s
vxi
vyi
= 50°
![Page 17: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/17.jpg)
Analyzing Motion in the x and y directions independently.
x-direction: ddxx = v = vix ix t = vt = vfxfxtt vvixix = v = viicoscos
y-direction:y-direction: ddyy = ½ (v = ½ (vii + v + vff) t = v) t = vavgavg t t vvff = v = viyiy + gt + gt ddyy = v = viyiy t + ½ g(t) t + ½ g(t)22
vfy2 = viy
2 + 2gdd vviyiy = v = viisinsin
![Page 18: Motion in Two Dimensions Chapter 7.2 Projectile Motion What is the path of a projectile as it moves through the air? Parabolic? Straight up and down?](https://reader036.vdocument.in/reader036/viewer/2022062722/56649f285503460f94c4051e/html5/thumbnails/18.jpg)
Key Ideas
Projectile Motion: Gravity is the only force acting on a
projectile. Choose a coordinate axis that where
the x-direction is along the horizontal and the y-direction is vertical.
Solve the x and y components separately.
If time is found for one dimension, it is also known for the other dimension.