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### Kinematics Equations in Two Dimensions

I.A.1 Kinematics in One Dimension

- Mechanics – motion and the forces that cause that motion
- Kinematics – describes motion without regard to the forces that cause that motion
- Dynamics – describes the forces that cause the motion

Ex. A kitten is thrown straight upward from the edge of a cliff the is 30 m high. The kitten rises 10 m and then falls all the down to the base of the cliff. What is the distance the kitten travels? What is the displacement of the kitten?

Example: The initial position of a runner is 50.0 m. 3.00 s later, the runner is at 30.0 m. What is the average velocity of the runner?

Acceleration (also known as “what’s a meter per second per second?”)

A brief and simple, yet fundamentally important comparison of velocity and acceleration.

Example: A car is traveling in a straight line along a highway at a constant speed of 80.0 km/h for 10 s. What is the acceleration of the car?

Example: During the time interval of 9.0 s to 14 s, a drag racer slows (using a parachute – or perhaps by dragging a comatose llama in a burlap bag) from 15.0 m/s to 5.0 m/s. What is the acceleration?

A few notes on signs and acceleration

- If acceleration and velocity have the same sign, the object is increasing in speed.
- If acceleration and velocity have opposite signs, the object is decreasing in speed.

A few light and humorous moments as Mr. Evans walks across the front of the room.

Concept Check: A car traveling with a constant speed travels once around a circular path. Which of the following is true regarding the car’s motion?

- The displacement is zero.
- The average speed is zero.
- The acceleration is zero.

Example: An object with an initial velocity of 4.0 m/s travels along a straight path with constant acceleration. In a time of 3.0 s, the object increases its velocity and travels a distance of 27 m. What is the final velocity of the object?

Ex. A race car starts from rest and accelerates at –5.00 m/s2. What is the velocity of the car after it has traveled –30.5 m?

Ex. A drag racer starting at x = 50.0 m accelerates from rest at a constant rate of 8.0 m/s2. a) How fast is the car going at t = 10.0 s? b) How far has it traveled at t = 10.0 s. c) What is the average velocity for the time interval from 0-10.0 s?

A couple of modifications to the kinematic equations

- Since displacement is vertical replace x with y
- a = g = –9.81 m/s2

Example: A stone is dropped from a tall building (this is against the law and very unsafe by the way). What is the vertical displacement of the stone after 4.00 s? What is its velocity at this point?

Example: A melon is thrown upward from the top of a tall building with an initial velocity of 20.0 m/s. Find the a) time for the melon to reach its maximum height b) the maximum height c) the time for the melon to return to the thrower d) the time when the melon is 22.5 m below its initial height.

Some notes on freely falling bodies.

- Object is not necessarily moving down, but g is downward
- Compare v and g for an object tossed upward
- An object launched upward and downward with same vo

Concept check: A baseball is thrown upward. What is the magnitude and direction of the ball’s acceleration:

- At one-half of it maximum height as the ball rises?
- At maximum height?
- At one-half of the maximum height as the ball falls?

Velocity vs. time graphs (slightly, although not intensely, more confusing)

The Slow, Painful Death of the AP Physics Student

This is important – the two velocity vectors in each case are independent of each other.

An object launched horizontally (an instructive and illustrative figure)

Concept check: Describe (magnitude and direction)

- The horizontal component of an object’s velocity if the object is launched horizontally
- The vertical component of an object’s velocity if the object is launched horizontally
- The speed of an object that is launched horizontally
- The acceleration for an object launched horizontally

Another concept check

- Graphing horizontal velocity and vertical velocity vs. time
- Graphing acceleration vs. time

Strategy: almost all of the AP problems you see involving projectile motion have time as a common factor for both horizontal and vertical components of motion.

Ex. An airplane moves horizontally with a constant velocity of 115 m/s at an altitude of 1050 m. A package is released that falls along a curved trajectory. How long does it take for the package to hit the ground? What is the horizontal displacement of the package from its initial release point?

Ex. A car driven by a deranged orangutan drives straight off the edge of a cliff that is 54 m. high. The police at the scene of the accident note that the point of impact was 130 m from the base of the cliff. How fast was the car moving when it went over the cliff?

Ex. A horizontal rifle is fired at a bull’s eye. The muzzle speed of the bullet is 670 m/s and the horizontal distance between the end of the rifle and the bull’s eye is 48 m. If the barrel of the rifle is horizontal when the bullet is fired, how far below the bull’s eye does the bullet strike the target?

Ex. An Alaskan rescue plane drops a package of Twinkies to a stranded party of explorers who have resorted to cannibalism. The plane is traveling horizontally at 40.0 m/s at a height of 100.0 m above the ground. What is the horizontal distance the package travels before it strikes the ground?

Ex. A baboon on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must the motorcycle leave the top of the cliff if it is to land on level ground below, 90.0 m from the base of the cliff?

Ex. A girl throws a pancreas horizontally with a speed of 10.0 m/s from a bridge. It travels a horizontal distance of 20.0 m before striking the water. From what height is the pancreas thrown?

An object launched at an angle above the horizontal (note the expression for the components)

Concept check: Describe (magnitude and direction)

- The horizontal component of an object’s velocity if the object is launched at an angle
- The vertical component of an object’s velocity if the object is launched at an angle
- The speed of an object that is launched at an angle
- The acceleration for an object launched at an angle

Another concept check

- Graphing velocity vs. time
- Graphing acceleration vs. time

A useful tidbit of information: in the absence of air resistance, an object launched at an angle above a horizontal surface achieves maximum displacement when launched at an angle of 45º.

Example: A long jumper leaves the ground at an angle of 20.0º to the horizontal at a speed of 11.0 m/s. a) How far does he jump if the landing pit is the same height as the board from which he jumps? b) What is the maximum height reached?

Ex. A place kicker must kick a football from a point 36.0 m from the goal. The ball must clear the crossbar which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53º with the horizontal. By how much does the ball clear or fall below the crossbar?

Ex. A daredevil is shot out of a cannon at an angle of 45.0º to the horizontal with an initial speed of 25.0 m/s. A net is positioned at a vertical height of 15.0 m above the cannon. At what horizontal distance should the net be placed to catch the daredevil?

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