1. Motion in Two Dimensions
by: Heather Britton

2. Motion in Two Dimensions
We have studied how motion takes place in terms of displacement, time, velocity, and acceleration in one dimension
Now we will look at what happens in two dimensions using the same variables and equations

3. Motion in Two Dimensions
Projectile motion - an object (projectile) moving freely in the air under the influence of gravity alone
We will not concern ourselves with how the object was projected just what happens afterward

4. Motion in Two Dimensions
Examples of projectiles include
A thrown ball
A speeding bullet
A person jumping
A skateboarder going off a curb

5. Motion in Two Dimensions
Solving problems involving 2D motion involves working with vectors
We will either add vectors to obtain a resultant
Or we will resolve vectors into components

6. Motion in Two Dimensions
When we work with components we will use the cartesian plane and x and y values
This will allow us to form right triangles and use trigonometric identities
Lets look at 3 different examples involving a flying plane

7. Motion in Two Dimensions
Example 1
What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South with a tail wind of 20 km/hr?

8. Motion in Two Dimensions
Example 2
What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South into a head wind of 20 km/hr

9. Motion in Two Dimensions
Example 3
What is the resultant velocity (magnitude and direction) of a plane flying at 100 km/hr South and a cross wind blowing due East at 20 km/hr?

10. Motion in Two Dimensions
dia/vectors/plane.cfm

11. Motion in Two Dimensions
Example 4
You can swim with a velocity of 4 m/s. What would your resultant velocity (magnitude and direction) be if you were swimming across a river with a current of 1 m/s?

12. Motion in Two Dimensions
When given a resultant velocity and an angle, you can find the component velocities (use x and y subscripts)
Using the cartesian plane draw your resultant velocity and angle
For projectile motion x is horizontal (adjacent) and y is vertical (opposite)

13. Motion in Two Dimensions
Example 5
A swimmer crosses a river with a resultant velocity of 6 m/s. Because of the current she lands at an angle of 35 degrees from the horizontal. How fast does she swim, and how fast is the current?

14. Motion in Two Dimensions
In projectile motion we can treat what happens in the x direction as a constant velocity problem
We can treat what happens in the y direction as a constant acceleration problem

15. Motion in Two Dimensions
Practice finding component velocities (vox and voy)
vo = 15 m/s θ = 0°
vo = 8 m/s θ = 30°
vo = 45 m/s θ = 62°

16. Motion in Two Dimensions
dia/vectors/pap.cfm

17. Motion in Two Dimensions
Since the velocity in the x direction is constant we can use the equation for constant velocity
vx = x / t
x represents the displacement in the x direction not the distance the projectile actually flew

18. Motion in Two Dimensions
The y component velocity is undergoing constant acceleration due to gravity
This means that vy is constantly changing
The vy is changing at the rate of 9.8 m/s2
So with constant acceleration in projectile motion a = g = 9.8 m/s2

19. Motion in Two Dimensions
The equations to be used for vy are
1. vy = voy + at
2. y = voyt + (1/2)at2
3. vy2 = voy2 + 2ay
4. avg vy = (vy + voy)/2

20. Motion in Two Dimensions
The y component velocity will determine the time an object spends in the air
The greater percentage of y component velocity the greater height the object will reach
Whenever time needs to be determined use the y component velocity

21. Motion in Two Dimensions
dia/vectors/nhlp.cfm
s/vectors/u3l2b.cfm

22. Motion in Two Dimensions
Example 6
A cliff diver must clear the rocks on the shore. If the rocks jut out 5.0 m and the cliff is 60.0 m high, what is the minimum horizontal velocity the diver must leave the ledge with?

23. Motion in Two Dimensions
Example 7
A cannon ball is launched at 40 degrees above the horizontal with an initial velocity of 50 m/s. (1.) What are the component velocities? (2.) How high does it go? (3.) How long is it in the air? And (4.) How far down range (x axis) does it land?

24. Motion in Two Dimensions
Example 8
A cannon ball is launched at 50 degrees above the horizontal with an initial velocity of 50 m/s. (1.) What are the component velocities? (2.) How high does it go? (3.) How long is it in the air? And (4.) How far down range (x axis) does it land?

25. Motion in Two Dimensions
Examples 7 and 8 show the significance of complementary angles
Complementary angles add up to 90°
For complementary angles the range (horizontal displacement) will be the same
The height attained and the time spent in the air will be different

26. Motion in Two Dimensions
There is one angle that will give the greatest range
That angle is 45°
Therefore maximum horizontal displacement can be obtained by launching a projectile at a 45° angle

27. Motion in Two Dimensions
Example 9
On a 130 m par 3 a ball is hit off the tee s later it lands in the hole. What was the initial velocity and what angle was the ball hit with?