SPH3U1
Lesson 12
Kinematics
PROJECTILE MOTION
LEARNING GOALS
Students will:
•
•
•
Describe the motion of an object thrown at arbitrary angles through the air.
Describe the horizontal and vertical motions of a projectile.
Solve projectile motion problems.
PATH OF A PROJECTILE
1.
Logon to a computer. Go to the website
http://www.walter-fendt.de/ph14e/projectile.htm
Change the initial height to 0. Run the simulation.
a)
What is the shape of the path followed by the projectile?
b)
Try different angles of inclination. Which one gives you the maximum distance?
c)
What other angle gives you the same distance as 300? As 550?
d)
What is the rule for finding two angles that give the same distance?
VELOCITY OF A PROJECTILE
2.
Click on the button to show velocity. Also click on the box for Slow Motion. Run the
simulation. You will see the overall velocity vector as well its horizontal and vertical
components.
a)
Focus on the horizontal component only. Run the simulation. What does it show
for the horizontal velocity? Why do you think this is happening?
b)
Focus on the vertical velocity component. Run the simulation. Describe what is
happening to the vertical velocity component. What is causing this?
c)
Focus on the overall velocity. Does it ever become zero? Why or why not?
d)
Look at how the overall velocity is oriented to the path of the projectile. What
word do we use to describe this relationship?
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e)
Lesson 12
If you increase the initial velocity, describe what happens to the
Kinematics
distance travelled:
height reached:
ACCELERATION OF A PROJECTILE
3.
Click on the acceleration button. Run the simulation.
a)
What can you say about the acceleration of the projectile? What causes it to be
like this?
b)
Why is there no horizontal component to the acceleration?
SOLVING PROJECTILE MOTION PROBLEMS
IMPORTANT RULE: The horizontal part of the motion is independent from the vertical
motion. The two parts can be solved separately. HOWEVER, the TIME to complete the
horizontal motion is the same as the TIME to complete the vertical motion.
CONCEPT PROBLEMS
5.
A student pushes a coin off the edge of a counter with some horizontal velocity. At the
instant the coin goes over the edge, a second student drops a second coin from countertop level so it falls vertically to the floor. Which coin hits the floor first? Explain your
answer. Think about the IMPORTANT RULE above. TRY IT.
6.
A similar experiment to the one above is done. A rifle is fired horizontally over a large flat
plane. At the instant the rifle is fired, a student drops a second identical bullet from the
same height as the rifle. Which bullet hits the ground first? Explain.
7.
A monkey is hanging from a tree branch. A scientist wants to shoot the monkey with a
tranquilizer dart so she can study the monkey. She knows the monkey will be startled by
the sound of the tranquilizer gun and will let go of the branch at the instant the gun is
fired. Where should the scientist aim to hit the monkey? (Above it, below it or directly at
it?) Explain.
To check this answer, go to youtube.com and search for “MIT Physics Monkey and Gun”.
Go to full screen mode. Pause the video just before the projectile is launched. Use the
side of a piece of paper placed along the edge of the gun to see where it is aimed.
Remove the paper and continue watching the video. Fix your answer above if necessary.
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Lesson 12
Kinematics
EXAMPLE
A ball is rolled at 2.2 m/s along a counter top that is 1.2 m above the floor. It rolls over the
edge of the counter top.
a)
How far horizontally does it travel before hitting the floor?
b)
What is the velocity of the ball when it hits the floor?
SPLIT THE GIVENS INTO HORIZONTAL AND VERTICAL QUANTITIES
Quantity
Horizontal (x)
Vertical (y)
let up be +
initial velocity
2.2 m/s
final velocity
2.2 m/s
acceleration
0
0
-9.8 m/s2
displacement
-1.2 m
Vertical: The acceleration is the
acceleration due to gravity. The negative
sign means down. The displacement is
also down as the ball starts vertically at
the counter height and ends vertically on
the floor which is lower.
Time: is the same for both x and y.
time
a)
Horizontal: The acceleration is 0 so the
velocity is constant.
Now you can solve the problem as normal. You have enough information to solve
for the remaining vertical quantities. Solve for time:
Vertical:
∆
+ ∆ ∆ = −1.2 = 0 + −9.8 ∆ ∆ = 0.495
Horizontal: solve for distance – since v is constant
∆ = ∆ = 2.2 0.495 = 1.09
b)
The final velocity has two components. We already know the final horizontal
velocity (2.2 m/s). Now find the final vertical velocity.
Vertical:
+ ∆
= = 0 + −9.8 0.495 = −4.85
Combining:
= 2.2
+ 4.85
= 5.33/
"# =
$.%&
.
∴ # = 66)
therefore the velocity of the ball on contact
2.2 m/s
θ
4.85 m/s
with the floor is 5.3 m/s [660 below horizontal].
HOMEWORK
Do P78 Q1-2, P81 Q1-2 (Top) Q2-8 (Bottom)
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SPH3U1
Lesson 13
Kinematics
PROJECTILE MOTION II
LEARNING GOALS
•
•
•
Students will describe the motion of an object thrown at arbitrary angles through the air
Students will describe the horizontal and vertical motions of a projectile
Students will solve projectile motion problems
PATH OF A PROJECTILE
Logon to a computer. Go to the website
http://www.walter-fendt.de/ph14e/projectile.htm
All the examples you did in the previous lesson had either the initial horizontal or the initial
vertical velocity equal to zero. Or, the two were separately specified. Click on Slow motion and
on the velocity button. Run the simulation.
The thickest, darkest velocity vector is the overall velocity. You will now solve a problem where
this is the only initial velocity you are given.
EXAMPLE
1.
A golfer hits a golf ball with an initial velocity of 54 m/s [420 above the horizontal]. The
ball lands on an elevated green that is 14 m higher that the level from which the golfer hit
the ball.
a)
How far away horizontally from the golfer does the ball land?
b)
What is the velocity of the ball when it lands?
You will solve this example using what you already know with some guidance. Fill in the
quantities you know in the table. For the initial vertical and horizontal velocities, take the
velocity given above and determine its components using trigonometry.
Quantity
Horizontal (x)
Vertical (y)
let up be +
initial velocity
final velocity
acceleration
0
-9.8 m/s2
displacement
time
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a)
Lesson 13
Kinematics
Now solve the vertical problem to determine the time. This will produce a
quadratic equation that will give you two answers. If you do it right, you will get
the times 0.4 s and 7.0 s.
Which of the two times is the one you want to use? Why?
Now determine the horizontal distance the ball travels. You should get 279 m.
b)
Find the final vertical velocity (32 m/s [down]). Add it to the final horizontal
velocity (40 m/s) to get the final answer (51 m/s [390 below the horizontal]).
HOMEWORK
Do p 92 #2-4
p113 #27, 29, 30, 32, 33
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SPH3U Physics
Kinematics
Projectile Problem Solving
As we learned yesterday when an object moves freely under the influence of gravity it accelerates
in the downward direction only. The horizontal components of the velocity remain constant.
Motion in the horizontal direction can be described by
Motion in the vertical direction can be described by the 5 equations of motion.
Solve the following question in your groups using the whiteboards:
An arrow is shot from a height of 20.0m with an initial horizontal velocity of 18.0 m/s.
a.
Sketch (Must be checked by teacher before moving on)
b.
How far in the horizontal direction will the arrow fall?
SPH3U Physics
Kinematics
Ramp & Cup Projectile Test
Group Members:
___________________________________
___________________________________
___________________________________
Purpose: To analyze the motion of a horizontally launched projectile.
Setup:
Data (from teacher):
Height of lab bench: __________
Average horizontal displacement of fall: ____________
Calculations:
1. Determine the time the ball takes to fall.
2. Determine the projectile’s horizontal velocity.
The Test:
To test the power of predictability your teacher will tape a cup to a retort stand at a certain height
above the ground. Using your results above you must calculate where the retort stand must be
placed on the floor so the marble lands in the cup (ie. Find the horizontal displacement between
edge of table and middle of cup). Complete a full formal solution. Once you have done the
calculation bring it to your teacher to test your prediction.
Height of cup for your group (obtain from teacher): _________________
Based on your calculations, where should the cup be placed? _________________
To be completed by teacher: The marble landed
in
near
far from
the cup