1. Kinematics in 2-D

2. Review - What is Kinematics???
Describes the motion of objects
Uses a set of equations
Draws a relationship between time, distance, velocity, and acceleration

3. We Started in 1-D One dimensional motion (1-D)
Motion only in one plane (either in the x-direction or the y-direction)
Motion in a straight line

4. Now Let’s Do it in 2-D Two Dimensional Motion
Instead of motion in either the x-direction or the y-direction, we have motion in both the x-direction and the y-direction
The motion is no longer in a straight line; it is parabolic

6. The Projectile
Any object that travels through the air in which the only force acting on it is gravity

7. What are some example of projectiles?

8. And another…

9. Some Important Things to Know
Projectiles travel in 2 dimensions, in the x-direction AND in the y-direction
The ONLY force acting on a projectile is gravity
The motion in the x-direction is COMPLETELY INDEPENDENT of the motion in the y-direction…let’s take a look

14. One More Look at Some Projectiles

15. Sign Conventions Positive directions (+d) Negative directions (-d)
Right Up
Negative directions (-d)
Left
Down
Now it is important as ever!!!

16. The Good News…
The same equations from 1-D kinematics apply in 2-D

17. Solving Problems Draw a picture of the problem
Define your sign convention
It is your decision, but use one that is convenient
Typically, up is positive and right is positive
Separate the x-components and the y-components so you can solve them separately
Use the GUESS method

18. Remember g?
Earlier, we said g for the surface of the Earth equals 9.81 m/s2
We were not taking into consideration the direction in which g works
What do you think g is?

19. Crazy Important…
The motion in the x-direction is COMPLETELY INDEPENDENT of the motion in the y-direction
There is one link between the x-direction and the y-direction, and it is t

20. Lingo and Such Terms Variables/Constants Projectile Trajectory Range
Altitude
Hang time
Vi,x or V0,x
Vi,y or V0,y
Vf,x
Vf,y t
d or h g

21. Example Problem
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

22. Step 1: Draw a Picture
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

24. Step 1: Draw a Picture
Step 2: Define a Sign Convention
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

26. Step 1: Draw a Picture
Step 2: Define a Sign Convention
Step 3: Pick the direction you want to start in and solve!
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?

27. Let’s Start in the Y Direction
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?
Given: d = m g = -9.8 m/s2 Unknown: t = ? Equation:

28. Let’s Start in the Y Direction
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?
Substitute: Solve:

29. to solve the X component
Now we have what we need
to solve the X component

30. Now Let’s Solve the X-Direction
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?
Given: vi,x = +115 m/s t = 14.6 sec Unknown: dx = ? Equation:

31. Now Let’s Solve the X-Direction
An airplane is flying with a constant velocity of 115 m/s. It drops a package from an altitude of 1050 m. If we ignore air resistance, how far down range will the object be when it hits the ground?
Substitute: Solve:

32. 1679 meters

33. Practice #1
A car drives off a cliff that is 50 m high. When it lands, investigators measure that the car is 85 m away from the base of the cliff. Calculate the following:
the time the car was in the air.
the horizontal velocity of the car when it drove off the cliff?

34. Practice #2
You throw a ball straight up with an initial vertical velocity of 22 m/s. Calculate the following:
the time the ball is in the air.
how high the ball goes.