PHYS 201
Chapter 3 Kinematics in 2-D
Equations in 2-D
Projectile
Lab This Week: Forces and Equilibrium
•
•
•
Equilibrium: forces balanced
Weight: Force due to gravity: Fg = mass*g
Balance forces: both in x and y
R
A
B
C
F
F
x
0
y
0
Equations with Constant Acceleration in 2D
Note: Near the Earth surface, ‘ay = g’.
X
y
vx = v0x + axt
vy = v0y + ayt
x = ½ (v0x + vx)t
y = ½ (v0y + vy)t
x = v0xt + ½ axt2
y = v0yt + ½ ayt2
vx2 = v0x2 + 2 axx
vy2 = v0y2 + 2 ayy
Projectile Motion
If an object is launched near Earth (or a planet) surface, then it follows a parabolic path.
It is called the projectile motion.
E.g. - a bullet fired from a gun
- a golf ball
- a cargo drops from a plane
Note: A rocket generally does not follow a projectile motion because it has its own engine to produce a
thrust.
Projectile Motion
Following factors are concerned:
How high it travels? (in vertical direction)
How far it travels ? (in horizontal direction)
How long it takes to travels ? (the same for both vertical and horizontal directions).
Projectile Motion
The horizontal (x) and vertical (y) motions are independent.
The horizontal motion has a constant velocity (ax = 0)
The vertical motion has a constant acceleration due to gravity (g = -9.81 m/s2)
Projectile Motion
The horizontal (x) and vertical (y) motions are independent.
The horizontal motion has a constant velocity (ax = 0)
The vertical motion has a constant acceleration due to gravity (g = 9.8 m/s2)
Projectile Motion
Cannon
http://phet.colorado.edu/en/simulation/projectile-motion
CLICKER!
Ball ‘A’ is dropped from rest. Ball ‘B’ is shot horizontally from the same height and
time as ‘A’. Which ball will hit the floor first?
1.
2.
3.
Ball A
Ball B
Both hit at the same time
Vertical motion is independent of horizontal motion
SHOOTER-DROPPER DEMONSTRATION
Link: https://shibboleth.ohio.edu/shibbolethidp/SSO?colcode=ou&brcode=main&shire=https%3A%2F%2Fstreamer.ohiolink.edu%2FShibboleth.sso%2FSAML%2FPOST&time=1284882525&target=co
okie%3Acd9a85c7&providerId=https%3A%2F%2Fstreamer.ohiolink.edu%2Fshibboleth&
CLICKER!
A cart travels at constant speed along a level track. At a given point, a ball pops
straight up from the cart. Which of the following is the best option for the behavior of
the ball.
1.
2.
3.
4.
The ball will land in front of the cart
The ball will land in the cart
The ball will land behind the cart
The ball at the point on the track where the ball was launched
Horizontal motion is independent of vertical motion.
In this case, both objects have constant horizontal velocity
CLICKER!
A soccer ball is kicked on a level playing field. It's initial velocity is 20m/s at an angle
of 30° above the horizontal.
Which of the following is the most accurate statement about the speed of the ball?
1.
2.
3.
4.
5.
it is a minimum at maximum height
it is zero at maximum height
it is a minimum just before it lands
it is a minimum just after it is kicked
it has the same speed throughout the trajectory
vx constant. Speed depends on both vx and vy. Smallest speed when vy=0.
http://capa2.phy.ohiou.edu/res/ohiou/physlets/2dkinematics/proj2.html
Ex. 1
A soccer ball is kicked at an initial speed of 20.0 m/s at 30º angle above the horizontal from
ground.
a). What are the horizontal and vertical components of the initial velocity?
b). How high the ball will reach?
c). How long it takes to reach the ground?
d). How far it travels?
e). What is the speed of the ball at the maximum height?
CLICKER!
A red ball is shot vertically with an initial velocity of 20 m/s. At the same time, a green ball is
thrown at an angle of 30 deg. Its vertical component of initial velocity is also 20 m/s.
Which ball will arrive to the highest height first?
1). A
2). B
3). Both at the same time.
Ex 2. Flight time.
A red ball is shot vertically with an initial velocity of 19.62 m/s. At the same time, a green ball is
thrown at an angle of 60 deg. Its vertical component of initial velocity is also 19.62 m/s.
Find the time to reach
a) maximum height.
b) to the ground.
.
Ex 3. Vertical height.
A red ball is shot vertically with an initial velocity of 19.62 m/s. At the same time, a green ball is
thrown at an angle of 60 deg. Its vertical component of initial velocity is also 19.62 m/s.
How high it will be at 1 s?
.
Ex 4.
A red ball is shot vertically with an initial velocity of 19.62 m/s. At the same time, a green ball is
thrown at an angle of 60 deg. Its vertical component of initial velocity is also 19.62 m/s.
How long it will take to reach 14.72 m height?
Dy = voy t + ½ ayt2
14.72 = 19.52 t – ½ * 9.81 t2
3 = 4t – t2
t2 -4t +3 = 0. ---(*)
Dy = 14.72 m
v0y = 19.52 m/s
ay = -9.81 m/s2
CLICKER!
The diagram below shows the motion diagram of a soccer ball. The images are left
at 1 second intervals. Ignore air resistance.
At which point is the magnitude (size) of the horizontal component of the velocity
the greatest? (point (4) represents the maximum height)
(3)
(2)
(1)
(4)
(5)
(8) Points 1 and 7
(6)
(7)
(9) All Points Same
CLICKER!
How does the magnitude of the vertical component of the velocity compare between
points A and B?
(1) Points A Greater
(A)
(2) Point B Greater
(B)
(3) Both Same
At a fixed height magnitude of velocity as it comes down is the same as when it was
going up. Directions are different.
Projectile
* Solve ‘x’ and ‘y’ components separately. The ‘x’ and ‘y’ components can be linked via the
time, ‘t’, which is the same for both axes.
* Dy can be ‘-’ or, ‘+’ or, ‘0’.
* vx = v0x = v0 cosq = constant
•At the top of the path, vy = 0.
•Upward path, vy  +.
•Downward path, vy  -.
Ex. 5
A soccer ball is kicked at an initial speed of 20.0 m/s at 30º angle above the horizontal from
ground.
a). What is the final speed of the ball just before it hits the ground?
b). What is the height of the ball at t = 0.8 s?
c). What is the speed of the ball at t=0.8s?
X
y
vx = Dx / t
vy = v0y + ayt
Dy = ½ (v0y + vy)t
Dy = v0yt + ½ ayt2
vy2 = v0y2 + 2 ayDy
Ex. 6
A firefighter is spraying water on a building. Water leaves the hose at 35 m/s and an angle of
30° above the horizontal, and hits the roof top of the building. The nozzle of the hose is 1.0m
above the ground, and the building is 22m away.
a). How much time is required the water to reach the roof top of the building?
b). How high is the roof of the building from the ground?
c). In what vertical direction is the water traveling when it hits the building?
d). What is the speed of the water just before it hits the building?
X
y
vx = Dx / t
vy = v0y + ayt
Dy = ½ (v0y + vy)t
Dy = v0yt + ½ ayt2
vy2 = v0y2 + 2 ayDy
Projectile
* Solve ‘x’ and ‘y’ components separately. The ‘x’ and ‘y’ components can be linked via the
time, ‘t’, which is the same for both axes.
* Dy can be ‘-’ or, ‘+’ or, ‘0’.
* vx = v0x = v0 cosq = constant
Ex. 7
A water balloon is launched at a speed of 11 m/s from ground level at an angle of 50 degrees
above the horizontal toward an innocent bystander who is 1.5m tall and standing 2m away.
a)
b)
By how much does the water balloon clear the person's head?
How fast is the balloon traveling when it is directly over the person's head?
If you know ‘t’, you can find ‘Dy’.
You can find ‘t’ from horizontal axis.
X
y
vx = Dx / t
vy = v0y + ayt
Dy = ½ (v0y + vy)t
Dy = v0yt + ½ ayt2
vy2 = v0y2 + 2 ayDy
Ex. 8
You kick a soccer ball on a distant planet where the acceleration due to gravity is twice that on
Earth. Your ball has the same initial velocity (angle and speed) on the two planets.
Is flight time for the ball on the planet shorter or longer than Earth?
X
y
vx = Dx / t
vy = v0y + ayt
Dy = ½ (v0y + vy)t
Dy = v0yt + ½ ayt2
vy2 = v0y2 + 2 ayDy
Ex. 9
In shooter dropper experiment, a steel ball is shot horizontally from 2 m above the ground. If
the initial speed of the ball is 20 m/s,
a) how long it takes the ball to reach the ground?
b) where it will hit the ground (how far it travels)?
c) what is its final speed just before it hits the ground?
X
y
vx = Dx / t
vy = v0y + ayt
Dy = ½ (v0y + vy)t
Dy = v0yt + ½ ayt2
vy2 = v0y2 + 2 ayDy
Launch point is at the top of parabolic curve. So v0y = 0, and v0 = v0x.
Other 2-D Kinematics Problems
Ex 10
A plane is landing at a speed of 50m/s and descending at an angle of 20 deg. The plane is
1000m from the runway. As part of a stunt for a movie, a jeep wants to travel directly
underneath the plane all the way to the runway.
What speed will the jeep need to travel?
The ‘x’ component of velocity is the velocity that the jeep needs to travel.
Note: The speed = the magnitude of velocity.
Other 2-D Kinematics Problems
Ex 11
A car is traveling 50 mi/hr in a direction 60º North of East. How long will it take to travel 100 mi
East?
50 mi/hr
N
W
E
S
60º
100 miles