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KINEMATICS IN TWO DIMENSIONS
๏ฟฝโ๏ฟฝ
๐๐
Recall: Polar to Rectangular๐=๐ฌ๐๐จ๐ฌ ๐ฝ
๐ ๐=๐ฌ๐ฌ๐ข๐ง๐ฝ
๐=โ๐โ๐+๐โ
๐
To find if and are given:
๏ฟฝโ๏ฟฝ ๐ ๐
๐ฝ=๐ญ๐๐งโ๐|๐จ๐ฉ๐ฉ๐จ๐ฌ๐ข๐ญ๐๐๐๐ฃ๐๐๐๐ง๐ญ|
Projectile MotionA projectile is a particle moving near the Earthโs surface under the influence of its weight only (directed downward). Projectile motion is a combination of horizontal and vertical motions.
a = gW
WW
Consider Projectile at an Angle:A red ball is projected at an angle q. At the same time, a yellow ball is thrown vertically upward and a green ball rolls horizontally (no friction).
A red ball is projected at an angle q. At the same time, a yellow ball is thrown vertically upward and a green ball rolls horizontally (no friction).
Note vertical and horizontal motions of balls
q
voy
vox
vo
Calculations for Projectile Motion (No Air Resistance)
Horizontal velocity:
Vertical velocity:
โข Horizontal velocity is constant (no net force along horizontal)
๐ ๐=๐๐๐๐๐๐=๐๐ cos๐ฝ
โข Vertical velocity varies (net force is the gravitational force/weight)
Kinematics equations for constant acceleration
๐๐๐=๐๐sin ๐ฝ
๐ ๐=โ ๐๐
Vertical and Horizontal Motion
Simultaneously dropping a yellow ball and projecting a red ball horizontally.
Why do they strike the ground at the same
time?
Once motion has begun, the downward weight is the only force on each ball.
Once motion has begun, the downward weight is the only force on each ball.
W W
Consider Horizontal and Vertical Motion Separately:
Compare Displacements and Velocities
0 s0 s
1 svox 2 s 3 s
1 svy2 s
vx
vy3 s
vx
vy
Horizontal velocity doesnโt change.
Vertical velocity just like free fall.
vx
Calculation Range:Range is sometimes referred to the horizontal displacement for a complete projectile.
๐๐
๐ฃ๐
๐ฃ=๐ฃ๐ฅ
๐ฃ=๐ฃ๐
range,
โ ๐=๐๐
๐sin๐๐ฝ๐
- angle that can give maximum range๐ฝโฒ=๐๐๐จโ ๐ฝ
Example 1: A ball rolls off the top of a table 1.2 m high and lands on the floor at a horizontal distance of 2.0 m. What was the velocity as it left the table?
1.2 m2.0 m
Given: โ horizontal displacement
โ vertical displacement
Find: - initial velocity
Solutions: ๐ฃ๐=๐ฃ๐ฅ ๐ฃ ๐ฅ=โ ๐ฅ๐ก
โ ๐ฆ=๐ฃ๐๐ฆ๐ก+12๐๐ก 2 ๐ก=โ 2โ ๐ฆ๐
๐ฃ๐=โ ๐ฅ
โ 2โ ๐ฆ๐ ๐ฃ๐=โ ๐ฅโ ๐2โ ๐ฆ
๐ฃ๐=(2.0m ) โ 9.8m /s2
2 (1.2m )๐๐=๐ .๐๐ฆ /๐ฌ
Example 2:
The best leaper in the animal kingdom is the puma, which can jump to a height of 12.00 ft when leaving the ground at an angle of 45.0o. With what speed must the animal leave the ground to reach that height? How far has the puma leaped from its starting point?
Example 3:
Karl, who is a 2.00-m basketball player, is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0o angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket from the ground is 3.05 m.
Relative velocity
Relative velocity is all about relating measurements of two different observers,
one moving with respect to the other.
Measurements of velocity depend on the reference frame (coordinate systems) of the observers.
Problem Solving Strategy
1. Label each object involved with a letter that reminds you of what it is (for example, E for earth).
2. Look through the problem for phrase such as โThe velocity of A relative to B,โ and write the velocities as When a velocity is mentioned but it isnโt explicitly stated as relative to something, itโs always relative to earth.
3. Take the three velocities youโve found and assemble them into an equation: .
4. There will be two unknown components. Solve for them with the x- and y-components of the equation developed in step 3.
Example 1:
The boat is heading due north as it crosses a wide river with a velocity of 10.0 km/h relative to the water. The river has a uniform velocity of 5.00 km/h due east. Determine the velocity of the boat with respect to an observer on the riverbank.
Example 2:
A train is traveling with a speed of 15.0 m/s relative to earth. A passenger standing at the rear of the train pitches a baseball with a speed of 15.0 m/s relative to the train off the back end, in the direction opposite the motion of the train. What is the velocity of the baseball relative to earth?