35
2-D Motion Because life is not in 1-D
2-D MotionBecause life is not in 1-D
General
Solving 2-D Problems Resolve all vectors into components
x-component Y-component
Work the problem as two one-dimensional problems. Each dimension can obey different
equations of motion. Re-combine the results for the two
components at the end of the problem.
A roller coaster rolls down a 20o incline with an acceleration of 5.0 m/s2.
How far horizontally has the coaster traveled in 10 seconds?
How far vertically has the coaster traveled in 10 seconds?
A particle passes through the origin with a speed of 6.2 m/s traveling along the y axis. If the particle accelerates in the negative x direction at 4.4 m/s2
.
a. What are the x and y positions at 5.0 s?
A particle passes through the origin with a speed of 6.2 m/s traveling along the y axis. If the particle accelerates in the negative x direction at 4.4 m/s2
.
b. What are the x and y components of velocity at this time?
Projectile MotionMoves both horizontally and vertically, subject to acceleration by gravity in vertical direction.
Projectile MotionFired, thrown, shot, or hurled near the earth’s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored.
Horizontal Component of Velocity
Vertical Component of Velocity Undergoes accelerated motion Accelerated by gravity (9.8 m/s2 down)
Vy = Vo,y - gty = yo + Vo,yt - 1/2gt2
vy2 = Vo,y
2 - 2g(y – yo)
Horizontal and Vertical
Horizontal and Vertical
Zero Launch angle
A zero launch angle implies a perfectly horizontal launch.
vo
The Zambezi River flows over Victoria Falls in Africa. The falls are approximately 108 m high. If the river is flowing horizontally at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in freefall as it drops.
An astronaut on the planet Zircon tosses a rock horizontally with a speed of 6.75 m/s. The rock falls a distance of 1.20 m and lands a horizontal distance of 8.95 m from the astronaut. What is the acceleration due to gravity on Zircon?
General launch angle
vo
Projectile motion is more complicated when the launch angle is not straight up or down (90o or –90o), or perfectly horizontal (0o).
General launch angle
vo
You must begin problems like this by resolving the velocity vector into its components.
Resolving the velocityUse speed and the launch angle to find horizontal and vertical velocity components
VoVo,y = Vo sin
Vo,x = Vo cos
Resolving the velocityThen proceed to work problems just like you did with the zero launch angle problems.
VoVo,y = Vo sin
Vo,x = Vo cos
A soccer ball is kicked with a speed of 9.50 m/s at an angle of 25o above the horizontal. If the ball lands at the same level from which is was kicked, how far from the kicker does it land?
Trajectory of a 2-D Projectile
x
y
Mathematically, the path is defined by a parabola.
Trajectory of a 2-D Projectile
x
y
For a projectile launched over level ground, the symmetry is apparent.
Range of a 2-D Projectile
x
y
Range
Definition: The RANGE of the projectile is how far it travels horizontally.
Maximum height of a projectile
x
y
Range
MaximumHeight
The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.
Maximum height of a projectile
x
y
Range
MaximumHeight
The vertical velocity component is zero at maximum height.
Maximum height of a projectile
x
y
Range
MaximumHeight
For a projectile launched over level ground, the maximum height occurs halfway through the flight of the projectile.
Acceleration of a projectile
g
g
g
g
g
x
y
Acceleration points down at 9.8 m/s2 for the entire trajectory of all projectiles.
Velocity of a projectile
vo
vf
v
v
v
x
y
Velocity is tangent to the path for the entire trajectory.
Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
The velocity can be resolved into components all along its path.
Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
Notice how the vertical velocity changes while the horizontal velocity remains constant.
Velocity of a projectile
vy
vx
vx
vy
vx
vy
vx
x
y
vx
vy
Maximum speed is attained at the beginning, and again at the end, of the trajectory if the projectile is launched over level ground.
vo -
vo
Velocity of a projectile
Launch angle is symmetric with landing angle for a projectile launched over level ground.
to = 0
tTime of flight for a projectile
The projectile spends half its time traveling upward…
Time of flight for a projectile
to = 0
t
2t … and the other half traveling down.
Snowballs are thrown with a speed of 13m/s from a roof 7.0 m above the ground. Snowball A is thrown straight downward; snowball B is thrown in a direction 25o above the horizontal. When the snowballs land, is the speed of A greater than, less than, or the same speed of B? Verify your answer by calculation of the landing speed of both snowballs.
