1. Projectile Motion
Physics Level

2. The Motion Formulas
Good News/Bad News: These are the same formulas we used for linear motion. Do you know them?
If the answer is “NO”, then memorize them NOW!

3. Things we know to be true about all projectiles.
We assume NO AIR RESISTANCE! (Welcome to “Newtonia”), therefore…
The path of a projectile is a parabola.
Horizontal motion is constant velocity.
Vertical motion is in “free-fall”.
Vertical velocity at the top of the path is zero
Time is the same for both horizontal and vertical motions.

4. How are the formulas different for projectiles
How are the formulas different for projectiles? They must be applied along only one axis at a time.
horizontal or “x” – direction
vertical or “y” – direction
Remember that for projectiles, the horizontal (x) and vertical (y) motions must be separated and analyzed independently.

5. Projectiles Launched Horizontally
In addition to the values that are true for all
projectiles, the initial vertical velocity is zero.

6. A typical physics problem
A cannon ball is shot horizontally from a cliff. vx
What do we know?
For all projectiles…
height dy
And for horizontal projectiles…
Range, dx
Hint: You should always list your known values (preferably in the picture when applicable) at the beginning of any problem and assign those values variables.

7. Let’s put numbers on it. What do we want to find?
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Range, dx
Add the given values to our list of known values. Now that the diagram is drawn and labeled and we have identified and listed all of our “known” and “given” values for the problem, let’s begin by finding time.

8. Finding time of flight for a horizontally launched projectile.
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Range, dx
Since we know more values for vertical motion, let’s use it to find time. Start with the displacement equation (can use down as positive since all are down)…
Now solve for t…

9. Finding range (horizontal distance)
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Calculated:
Range, dx
Now that we know time, let’s find dx. Remember that horizontal motion is constant. Let’s use the displacement formula again but in the x direction.

10. Finding final velocity (magnitude and direction)
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Calculated:
Range, dx θ Vf
Final velocity requires a little more thought. Remember that velocity is a vector quantity so we must state our answer as a magnitude (speed that the projectile strikes the ground) and direction (angle the projectile strikes the ground). Also remember horizontal velocity is constant, therefore the projectile will never strike the ground exactly at 90°. That means we need to look at the horizontal (x) and vertical (y) components that make up the final velocity.

11. Finding final velocity (magnitude and direction)
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
Vxf = 5 m/s
height
Calculated:
Range, dx Vf
Let’s look more closely at the vector, vf. To help see it better, let’s exaggerate the angle. Since x- and y- motion are separate, there must be components.
So, we have the x-component already due to the fact that the horizontal velocity is constant. Before we can find vf, we must find the vertical component, vyf. θ
Vyf Vf

12. Finding final velocity (magnitude and direction)
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Calculated:
Range, dx θ Vf
To find vfy, remember that vertical motion is in “free-fall” so it is accelerated by gravity from zero to some value just before it hits the ground.
Calculating vyf:
Vxf = 5 m/s θ
Vfy Vf
Still not finished. Must put components together for final velocity.

13. Finding final velocity (magnitude and direction)
A cannon ball is shot horizontally at 5 m/s from a cliff that is 35 m high. Find time of flight, t = ? Find range, dx = ? Find final velocity, vf = ?
Vx= 5 m/s
Knowns:
Givens:
dy=35 m
height
Calculated:
Range, dx θ
Putting it together to calculate vf: Vf
Now we know both the x- and y- components of the final velocity vector. We need to put them together for magnitude and direction of final velocity.
Vxf = 5 m/s θ
Vfy=26.2 m/s Vf
Final Velocity = 26.7 m/s, 79.2° below horizontal

14. Projectiles Launched at an Angle

15. Separating the horizontal and vertical motions.vi θ
viy
vix
Since the initial velocity represents motion in both the horizontal (x) and vertical (y) directions at the same time, we cannot use it in any of our equations. Remember, the most important thing about projectiles is that we must treat the horizontal (x) and vertical (y) completely separate from each other. So…we need to separate “vi” into its x- and y- components. We will use the method we used for vectors.
Now that we have the components of the initial velocity, we will use only those for calculations. **Never use the original velocity at the angle in an equation! vi
vyi θ
vxi

16. Symmetry of the projectiles path.vf
vyf
vxf θ
vxf θ
vyf vf
A projectile’s path is a parabola, ALWAYS. That means if a projectile is launched and lands at the same height, there will be symmetry. So the angle of launch and angle of landing will be equal and the magnitude of the initial and final velocity will be equal. Therefore the magnitudes of the components will also be equal.
vxf
Since the vertical motion is the same as a ball that is thrown straight up (in free-fall), the initial and final y-components are equal but opposite. θ
Since the horizontal motion is always at constant velocity… vf
vyf

17. What do we know? Vy top = 0 vi Vyi dymax =height v xf = v xi θ Vxi vf
Range, dx vf
Vyf = -v yi
Knowns (for all projectiles):
Step 1: List known values! Draw and label picture.

18. Keeping the horizontal and vertical motions separate!
Vy top = 0 Vy
dymax =height vi θ Vx
Range, dx
Knowns (for all projectiles):
Step 2: Divide initial velocity into horizontal (x) and vertical (y) components.
Step 3: Find time if possible. Use vertical motion.

19. Finding time for projectiles that start and end at the same height
Finding time for projectiles that start and end at the same height. (Using distance formula.)
Almost every projectile problem can be solved by starting with the displacement equation to solve for time. In this case…
Vy top = 0 Vy
dymax =height vi θ
Note: There are three ways to find time for this problem. You may use any of them you wish.
Range, dx
Finding time – Method 1:
Since the initial and final vertical positions are both the same, vertical displacement dy = 0.
Now solve for time. Yes, it is a quadratic equation! This will be the time for the entire flight.
NOTE: If you want to find maximum height you will only use half the time. If you want to find range, use the total time.

20. Finding time for projectiles that start and end at the same height
Finding time for projectiles that start and end at the same height. (using symmetry)
Vy top = 0 vf θ vi θ Vy
vfy
dymax =height Vx
vfx
Range, dx
Find time – Method 2:
Use vertical motion and symmetry. Remember that the y component of initial and final velocities are equal and opposite. So using the formula
, plug in the known values from symmetry , vfy=-viy and solve for t.
Note: usually any vectors acting upward such as initial velocity are considered to be positive, therefore acceleration due to gravity is negative (-9.8) so the time will NOT turn out to be a negative.

21. Finding time for projectiles that start and end at the same height
Finding time for projectiles that start and end at the same height. (the zero vertical velocity at the top to find ½ the time.)
Vy top = 0 vf θ vi θ Vy
vfy
dymax =height Vx
vfx
Range, dx
Find time – Method 3:
Remember for all projectiles, the vertical velocity at the very top of the path is zero. If the projectile is launched and lands at the same height, the top of the path occurs at exactly the half-way point. If we use this as either the initial or final velocity we can calculate ½ the time of flight and then simply double it! Beginning with the same equation,
Remember to
use the symmetry between initial and final y-velocities!
To work from launch to top of path…
OR…To work from top of path to landing …
Remember: If you want hang time, double tup!

22. Let's do one with numbers.
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0
Vi= 30 m/s
60°
Vyi
dymax =height
Vxi
Range, dx
Knowns (for all projectiles):
Given values:
Step 1: List known values! Draw and label picture.

23. Let's do one with numbers.
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0
Vi= 30 m/s
60°
Viy
dymax =height Vx
Range, dx
Step 2: Divide initial velocity into x- & y- components.
Knowns (for all projectiles):
Given values:
We can add these to what we know. WE WILL NOT USE THE 30 m/s again in this problem because it is not purely an x- or y- value.

24. Finding time for projectiles that start and end at the same height
Finding time for projectiles that start and end at the same height. (Using distance formula.)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0
vy =26 m/s
dymax =height
vi= 30 m/s
60°
vix =15 m/s
Range, dx
Finding time – Method 1:
Remember that dy = 0 because the projectile is starting and ending at the same level (y-position). So, using the known and given values for this problem and the components we calculated, we can solve for time.
Since t = 0, could cancel a “t” or ignore that solution.

25. Finding time (using symmetry and initial and final velocities)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vfy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
Find time – Method 2:
Remember that the y component of initial and final velocities are equal and opposite. So, to calculate…
Note: This method gives the same answer as the previous one. Choose the one that make the most sense to you and use it.

26. Finding time t (using the zero vertical velocity at the top to find ½ the time.)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
Find time – Method 3:
Remember for all projectiles, the vertical velocity at the very top of the path is zero. If the projectile is launched and lands at the same height, the top of the path occurs at exactly the half-way point.
OR…To work from top of path to landing…
To work from launch to top of path…

27. Let's review what we know before we go on…
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ Vy
vfy
dymax =height Vx
vfx
Range, dx
Knowns (for all projectiles):
Givens:
Calculated values:
Now that I know time, I can add it to my list of known, given, and calculated values. To review…

28. Finding range (horizontal distance)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
Now that we know time, let’s calculate horizontal distance. Remember that horizontal acceleration is zero.

29. Finding maximum height (vertical distance)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
To find maximum height for this problem, remember that because it is launched and lands at the same level, maximum height occurs exactly half-way through the flight. So…USE ½ of the total time of flight.
Method 1:
Note: If you use the top as initial point and the end as final point, vyi = 0. Then dy = m but height = 34.5 m.

30. Finding maximum height (vertical distance)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
To find maximum height for this problem, you must use ½ of the total time of flight and final vertical velocity of 0 if initial point is at the beginning and final point is at the top.
Method 2:
Note: If you use the top as initial point and the end as final point, vyi = 0 and v yf = -26 m/s . Then dy = m but height = 34.5 m. 30

31. Finding maximum height (vertical distance)
A football is kicked from the ground with a speed of 30m/s at and angle of 60°. Find time of flight, range, and maximum height.
Vy top = 0 vf θ vi θ
vy =26 m/s
vy =-26 m/s
dymax =height
vx =15 m/s
vx =15 m/s
Range, dx
To find maximum height for this problem, you must use final vertical velocity of 0 if initial point is at the beginning and final point is at the top. Time is not needed for this equation!
Method 3:
Note: If you use the top as initial point and the end as final point, vyi = 0 and v yf = -26 m/s . Then dy = m but height = 34.5 m. 31

32. Projectiles that land higher than they are launched.vi dy θ dx
Vertical displacement is not zero. Consider the launch point as the zero height and then vertical displacement, dy, will be a positive number (as long as you continue to assume up is positive. Plug the value for dy into the above equation and solve for time. (Hint: Graph it and find the zeros! It’s easier than the quadratic equation.) The answer must be positive.
This will be the time for the flight to that point. Then you can use that time to find the horizontal distance the object traveled to get to that point.
NOTE: The highest point of the projectile DOES NOT occur at the half-way point of the flight. BE CAREFUL!

33. Projectiles that land lower than they are launched.
Again, vertical displacement is not zero. If the launch point as the zero height, vertical displacement, dy, will be a negative number (as long as you continue to assume up is positive. Plug the value for dy into the distance equation and solve for time. This will be the time for the flight to that point. Then you can use that time to find the horizontal distance the object traveled to get to that point.
NOTE: The highest point of the projectile DOES NOT occur at the half-way point of the flight. BE CAREFUL! vi θ
-dy dx 33

34. Hints for projectiles that are thrown downward
The key to this problem is drawing the picture and finding the components. Again, vertical displacement is negative along with the vertical component of the initial velocity and acceleration. Remember, to find final velocity, you must find the final y- velocity and then put the x- and y- components of the final velocity back together using the pythagorean theorem and the inverse tangent.
vix θi vi
viy
vfx
-dy vf
vfy θf dx
Some important notes:
The horizontal velocity remains the same (drawing shows otherwise but it is not to scale). The angle thrown downward is not the same as the angle it strikes the ground. Finally, the projectile will never strike perpendicular to the ground!