1. Projectile Motion or 2D Kinematics
By Joseph Calmer
References:
Conceptual Physics, Paul G. Hewitt, 10th edition, Addison Wesley publisher
l2a.html

2. Outline What is a projectile Characteristics of a projectile's motion
Horizontal and vertical components of velocity and displacement
Initial velocity components
Examples of problems

3. What is a projectile?
A projectile is an object upon which the only force acting is gravity. A projectile is any object which once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

4. Projectile Motion and Inertia

5. Horizontal and Vertical Velocities
A projectile is any object upon which the only force is gravity,
Projectiles travel with a parabolic trajectory due to the influence of gravity,
There are no horizontal forces acting upon projectiles and thus no horizontal acceleration,
The horizontal velocity of a projectile is constant (a never changing value),
There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down,
The vertical velocity of a projectile changes by 9.8 m/s each second,
The horizontal motion of a projectile is independent of its vertical motion.

6. Check for understanding
Write a 2 sentence summary
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7. Vector diagrams for projectile motion

8. Horizontal and vertical displacement – Horizontally Launched Projectile
If the horizontal displacement (x) of a projectile were represented by an equation, then that equation would be written as
x = vix • t
y = 0.5 • g • t2
(equation for vertical displacement for a horizontally launched projectile)‏

9. Displacement diagram of projectile motion
What is happening to
the vertical motion?
What is happening to
the horizontal motion?
Horizontal
Vertical

10. Horizontal and vertical displacement : Non - Horizontally Launched Projectile
y = viy • t • g • t2
(equation for vertical displacement for an angled-launched projectile)‏
If the horizontal displacement (x) of a projectile were represented by an equation, then that equation would be written as
x = vix • t

11. Check your understanding
1. Anna Litical drops a ball from rest from the top of 78.4-meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of motion?
2. A cannonball is launched horizontally from the top of an 78.4-meter high cliff. How much time will it take for the ball to reach the ground and at what height will the ball be after each second of travel?

12. Answers 1. It will take 4 seconds to fall 78.4 meters
Use the equation y = 0.5 • g • t2 and substitute -9.8 m/s/s for g. The vertical displacement must then be subtracted from the initial height of m.
At t = 1 s, y = 4.9 m (down) so height is 73.5 m (78.4 m m )‏
At t = 2 s, y = 19.6 m (down) so height is 58.8 m (78.4 m m )‏
At t = 3 s, y = 44.1 m (down) so height is 34.3 m (78.4 m - 45 m)‏
At t = 4 s, y = 78.4 m (down) so height is 0 m (78.4 m m)‏ 2.

13. Check your understanding
3. Fill in the table indicating the value of the horizontal and vertical components of velocity and acceleration for a projectile.
4. The diagram below shows the trajectory for a projectile launched non-horizontally from an elevated position on top of a cliff. The initial horizontal and vertical components of the velocity are 8 m/s and m/s respectively. Positions of the object at 1-second intervals are shown. Determine the horizontal and vertical velocities at each instant shown in the diagram.

14. Answers
3.The vx values will remain constant at 15.0 m/s for the entire 6 seconds; the ax values will be 0 m/s/s for the entire 6 seconds.
The vy values will be changing by -9.8 m/s each second. Thus,
vy = 29.4 m/s (t = 0 s) vy = 19.6 m/s (t = 1 s)‏
vy = 9.8 m/s (t = 2 s) vy = 0 m/s (t = 3 s)‏
vy = -9.8 m/s (t = 4 s) vy = m/s (t = 5 s)‏
vy = m/s (t = 6 s)‏
The ay values will be -9.8 m/s/s for the entire 6 seconds.
4.The vx values will remain 8 m/s for the entire 6 seconds.
The vy values will be changing by 9.8 m/s each second. Thus,
vy =9.8 m/s (t = 1 s) vy = 0 m/s (t = 2 s)‏
vy = -9.8 m/s (t = 3 s) vy = m/s (t = 4 s)‏
vy = m/s (t = 5 s) vy = m/s (t = 6 s)‏

15. Initial components of velocity
vx = v • cos α °
vy = v • sin α °

16. Evaluating various info
Determination of the Time of Flight
Determination of Horizontal Displacement
x = vix • t
Determination of the Peak Height
y = viy • t • g • t2

17. Equations of motion Horizontal motion Vertical Motion
the vertical acceleration of a projectile is known to be -9.8 m/s/s
Equations of motion
Horizontal motion
Vertical Motion

18. Solving Projectile Motion Problems
Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. For convenience sake, make a table with horizontal information on one side and vertical information on the other side.
Draw a picture of the situation- labling the knowns and unknowns
Identify the unknown quantity which the problem requests you to solve for.
Select either a horizontal or vertical equation to solve for the time of flight of the projectile.
With the time determined, use one of the other equations to solve for the unknown. (Usually, if a horizontal equation is used to solve for time, then a vertical equation can be used to solve for the final unknown quantity.)

19. Check your understanding
A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal displacement, and the peak height of the football.

20. Answer State the problem Use the appropriate equations of motion
The unknown quantities are the horizontal displacement, the time of flight, and the height of the football at its peak.
From the vertical information in the table above and the second equation listed among the vertical kinematic equations (vfy = viy + ay*t), it becomes obvious that the time of flight of the projectile can be determined. By substitution of known values, the equation takes the form of m/s = 17.7 m/s + (-9.8 m/s/s)•t
-35.4 m/s = (-9.8 m/s/s)•t
3.61 s = t
The total time of flight of the football is 3.61 seconds.
With the time determined, information in the table and the horizontal kinematic equations can be used to determine the horizontal displacement (x) of the projectile. The first equation (x = vix•t + 0.5•ax•t2) listed among the horizontal kinematic equations is suitable for determining x. With the equation selected, the physics problem once more becomes transformed into an algebra problem. By substitution of known values, the equation takes the form of x = (17.7 m/s)•( s) + 0.5•(0 m/s/s)•( s)2
x = (17.7 m/s)•( s)‏
x = 63.8 m
The horizontal displacement of the projectile is 63.8 m.
Finally, the problem statement asks for the height of the projectile at is peak. This is the same as asking, "what is the vertical displacement (y) of the projectile when it is halfway through its trajectory?" In other words, find y when t = 1.80 seconds (one-half of the total time). To determine the peak height of the projectile (y with t = 1.80 sec), the first equation (y = viy•t +0.5•ay•t2) listed among the vertical kinematic equations can be used. By substitution of known values into this equation, it takes the form of
y = (17.7 m/s)•(1.80 s) + 0.5*(-10 m/s/s)•(1.80 s)2
y = 31.9 m + (-15.9 m)‏
y = 15.9 m
The solution to the problem statement yields the following answers: the time of flight of the football is 3.61 s, the horizontal displacement of the football is 63.8 m, and the peak height of the football 15.9 m.

21. The Problem-Solving Approach
The following procedure summarizes the above problem-solving approach.
Use the given values of the initial velocity (the magnitude and the angle) to determine the horizontal and vertical components of the velocity (vix and viy).
Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. For convenience sake, make a table with horizontal information on one side and vertical information on the other side.
Identify the unknown quantity which the problem requests you to solve for.
Select either a horizontal or vertical equation to solve for the time of flight of the projectile. For non-horizontally launched projectiles, the second equation listed among the vertical equations (vfy = viy + ay*t) is usually the most useful equation.
With the time determined, use a horizontal equation (usually x = vix*t + 0.5*ax*t2 ) to determine the horizontal displacement of the projectile.
Finally, the peak height of the projectile can be found using a time value which one- half the total time of flight. The most useful equation for this is usually y = viy*t +0.5*ay*t2 .

22. A sample
A golfer practicing on a range with an elevated tee 4.9 m above the fairway is able to strike a ball so that it leaves the club with a horizontal velocity of 20 m s–1. (Assume the acceleration due to gravity is 9.80 m s–2, and the effects of air resistance may be ignored unless otherwise stated.)
a How long after the ball leaves the club will it land on the fairway?
b What horizontal distance will the ball travel before striking the fairway?
c What is the acceleration of the ball 0.5 s after being hit?
d Calculate the speed of the ball 0.80 s after it leaves the club.
e With what speed will the ball strike the ground?

23. 1. An object is projected horizontally at 8
1. An object is projected horizontally at 8.0 m/s from the top of a m cliff. How far from the base of the cliff will the object strike the ground?
2. An arrow is shot at 30.0° angle with the horizontal. It has a velocity of 49 m/s.
a. How high will it go?
b. What horizontal distance will the arrow travel?
3. A person kicks a rock off a cliff horizontally with a speed of 20 m/s. It takes 7.0 seconds to hit the ground, find:
a. height of the cliff
b. final vertical velocity
c. Range
4. A ship fires its guns with a speed of 400 m/s at an angle of 35° with the horizontal. Find the range and maximum altitude.
5. A basketball is held over head at a height of 2.4 m. The ball is lobbed to a teammate at 8 m/s at an angle of 40°. If the ball is caught at the same height it was tossed at, how far away is the teammate?

24. SUMMARY: create a sample problem for your neighbor to solve