1. Projectile motion

2. Equations Projectile motion:
The x-component equations depend on the x component of velocity, and the time.
The y-component equations depend on the initial y component of velocity, gravity, and the time.

3. Projectile motion
A projectile is an object in motion that is only affected by gravity.
Projectiles travel in trajectories: smooth curved paths that take the shape of a parabola.

4. Projectile motion
The range of a projectile is the total distance it travels before reaching the ground.
Can you identify the range in the picture below?

5. Projectile motion Range
The range is the total distance traveled along the x-axis. It equals x.
Range

6. Motion in the x and y directions
This figure shows the position of a projectile at equal time intervals.
What do you notice about the motion in the x direction? y x

7. Motion in the x and y directions
This figure shows the position of a projectile at equal time intervals.
What do you notice about the motion in the x direction?
The x-velocity is ________. y x

8. Motion in the x and y directions
This figure shows the position of a projectile at equal time intervals.
What do you notice about the motion in the x direction?
The x-velocity is constant.
In the y direction? y x

9. Motion in the x and y directions
This figure shows the position of a projectile at equal time intervals.
What do you notice about the motion in the x direction?
The x-velocity is constant.
In the y direction?
The y-velocity changes. It slows down, then speeds up. y x

12. Equations for projectile motion
Because horizontal and vertical are independant, there are two separate sets of equations for modeling projectile motion:
one set for the x axis
one set for the y axis

13. Equations for projectile motion
With x0 = 0 and ax = 0, the x-axis equations are:
Notice that vx is constant.
The projectile never speeds up or slows down in the x direction!

14. Equations for projectile motion
With x0 = 0 and ax = 0, the x-axis equations are:
Notice that vx is constant.
The projectile never speeds up or slows down in the x direction!
With y0 = 0 and ay = -g, the y-axis equations are:
These are just the equations for motion with constant acceleration, with a = g.

15. Understanding the subscripts
Splitting the motion into two sets of equations creates a lot of subscripts.
The subscript “y” or “x” tells you that the quantity relates to motion in the y or x direction.
For example: vy is the object’s velocity in the y direction.

16. Understanding the subscripts
Splitting the motion into two sets of equations creates a lot of subscripts.
The subscript of “0” tells you that this quantity is the starting value at t = 0 seconds.
For example: vy0 is the object’s velocity in the y direction at t = 0 s.

17. Projectile motion equations
Take another look at this set of equations.
x-axis equations: y-axis equations:
What variable do you see on BOTH the x-axis and y-axis?

18. Projectile motion equations
Take another look at this set of equations.
x-axis equations: y-axis equations:
What variable do you see on BOTH the x-axis and y-axis? t t t t
Time, t : motion in the x and y directions happens simultaneously!
Time is often the key to solving projectile motion problems.

19. Projectiles Launched at an Angle
A soccer ball kicked off the ground is also a projectile, but it starts with an initial velocity that has both vertical and horizontal components.
*The launch angle determines how the initial velocity divides between vertical (y) and horizontal (x) directions.

20. Steep Angle
A ball launched at a steep angle will have a large vertical velocity component and a small horizontal velocity.

21. Shallow Angle
A ball launched at a low angle will have a large horizontal velocity component and a small vertical one.

22. Range of a Projectile
The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle.
At what angle will the projectile reach the longest range?
What do you notice abut the ranges at angles 10*and 80*?
Do you see another example of this?
These are complimentary angles, when launched with the same initial velocity you will reach the same range.

23. Range of a Projectile
The vertical velocity is responsible for giving the projectile its "hang" time.

24. "Hang Time" You can easily calculate your own hang time.
Run toward a doorway and jump as high as you can, touching the wall or door frame.
Have someone watch to see exactly how high you reach.
Measure this distance with a meter stick.
The vertical distance formula can be rearranged to solve for time:

25. Projectile motion How do you use these equations to solve problems?
Let’s look at an example.
30 m/s

26. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
What is the initial velocity in the x direction? in the y direction?
30 m/s

27. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
What is the initial velocity in the x direction? in the y direction?
vx = 30 m/s vy0 = 0 m/s
30 m/s
Point out that since the velocity s constant in the x direction, no subscript is needed.

28. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
How far from the base of the cliff does the projectile land?
30 m/s
What variable are you being asked for?
Hint for the students: in the x-direction is the object moving with constant velocity or is it accelerating?

29. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
How far from the base of the cliff does the projectile land?
30 m/s
You are being asked for x.
60 m
Hint for the students: How far in the y-direction does the projectile go?

30. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
How high is the cliff?
30 m/s
What variable are you being asked for?
Hint for the students: How far in the y-direction does the projectile go?

31. Projectile motion
A projectile is fired horizontally off the top of a cliff with an initial velocity of 30 m/s. It hits the ground 2.0 seconds later.
How high is the cliff?
30 m/s
The projectile falls 19.6 m, so the cliff is 19.6 m high.
You are being asked for y.

32. Assessment
Which of the events described below cannot be an example of projectile motion?
a soccer ball kicked into the air
a car traveling down a hill
a rock thrown off a cliff
a package dropped from a plane

33. Assessment
Which of the events described below cannot be an example of projectile motion?
A boy on top of a roof has two balls. He throws one sideways at the same instant that he drops the second ball. Which ball hits the ground first?
a soccer ball kicked into the air
a car traveling down a hill
a rock thrown off a cliff
a package dropped from a plane

34. Assessment
Which of the events described below cannot be an example of projectile motion?
A boy on top of a roof has two balls. He throws one sideways at the same instant that he drops the second ball. Which ball hits the ground first? It’s a tie. Gravity pulls all objects down at the same rate!!!!!!!
a soccer ball kicked into the air
a car traveling down a hill
a rock thrown off a cliff
a package dropped from a plane

35. Assessment
A projectile is launched with an initial velocity of 13 m/s. The initial velocity components are vx0 = 5.0 m/s and vy0 = 12 m/s.
How long is it in the air?
Hint: use the y-axis equations to find t.

36. Assessment
A projectile is launched with an initial velocity of 13 m/s. The initial velocity components are vx0 = 5.0 m/s and vy0 = 12 m/s.
How long is it in the air?

37. Assessment
A projectile is launched with an initial velocity of 13 m/s. The initial velocity components are vx0 = 5.0 m/s and vy0 = 12 m/s.
What is the range of the projectile?