BIG-FIVE : please review - motion in one-dimension - see Kinematics LEVEL I

1)
A)To describe the motion of an object along a straight line and acted upon by a constant acceleration,
we need 5 fundamental quantities : __________________________

These 5 quantities are related by a group of five equations (the big five)
Can you find these five equations ?

		variable that's missing
five 1	d =	a (acceleration)
five 2	V =	d (displacement)
five 3	d =	V (Vf) (final velocity)
five 4	d =	Vi (initial velocity)
five 5	v2 =	T (time elapsed)

answers

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PROJECTILE MOTION: VOCABULARY
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2) Assumptions:
The pull due to gravity is the only (vertical) force acting of the projectile. we neglect air  resistance.
Gravity is directed downward. The surface of earth is a plane. Observe the picture below: \
WE SUPPOSE THE MOTION LASTS ABOUT 2 seconds.



source: the physics of every day phenomena. Mc Graw Hill.
We will work in the X,Y coordinate system .

A)
The projectile here is the ______________.
The trajectory is the path of the object. The trajectory is a  :
a) straight line  ?
 b) parabola ?  equation of a parabola ?
c) exponential ?

B)
The launch angle θ is ___________

C).
Vi is the initial velocity. Suppose the magnitude is 20  m/s. (about 45mph)
Vo has 2 components Vox = ________ along the X-axis (use trig functions to compute the x-component)
and Voy = ______________ along the Y-axis (use trig functions to compute the y-component)
D)
Let's call the maximum upward distance reached by the projectile H.
In the example above H =_____________ 

E)  Let's call the range or Xmax , the maximum horizontal distance traveled
Xmax = range = ________________ (loot at the image)

F)  Let's call Tmax the time to reach the maximum height. Tmax = ________
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V (or Vf in the big five) is the velocity of the projective at a given time t.
It also has 2 components.  (Vx, Vy) . 

(x,y)  is the coordinate of the projective at a given time. y = f(x) is the equation of the trajectory.

If Vi is given as well as the gravity, we can derive H, Tmax, the range or Xmax, V or (x,y) at any time. We will use the big fives and the big idea (Galileo Galilei 's idea)  :

To analyze the motion, separate the two-dimensional motion into
vertical and horizontal components.

The vertical motion is independent from the  horizontal motion
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3) Galileo found (through experiments, before Newton proved it using Calculus)
that a body in Free-Fall:
FAlls 1 unit of distance during the 1st second, 3 units of distance during the 2nd seconds
5 units  after the 3rd second ... This is the law of odd numbers.

A) CAn you complete the following picture? ( t=0s the projectile is at rest and starts to fall )
every 1 cm = 1 unit of distance.
Draw the ball at t=1s, t=2s, t=3s.


B) The total distance covered by the ball is therefore:
t = 1s, d = 1 unit
t = 2s , d = 4 units
t = 3 s, d = _____
t = 4s, d = _______
t = 5s, d = _______
Galileo found out that , in free-fall, the total distance covered by the object varies
as the _________ of the time.
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Newtons later derived the equation : d = 0.5 a t²  (a = -9.8m/s/s). (with Vi = 0m/s for t=0s)
showing that,  indeed, for an uniformly accelerated motion, the distance covered by the object
is proportional to the square of the time.

 If the ball is thrown with a horizontal velocity, it will still fall at the same rate as in Free-fall.
Observe the following image:


caption: 2 balls are thrown. One has no initial velocity and is a free-falling object.
One is thrown  with a horizontal velocity @ right and is a projectile.
THEY BOTH REACH THE GROUND AT THE SAME TIME.
ALONG THE VERTICAL THEY FALL AT THE SAME RATE. (y = 0.5 a t²)

Try to show that principle using experiments in class.
THese 2 experiments are shown in this movie:
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WORKSHEET

4) A) Suppose you want to shoot a Monkey in a tree. The Monkey stays still during the
shooting. How would you aim at it ? why ?
B.)Suppose now that the Monkey start falling as soon as he hears the detonation of your riffle.
How to aim now ? Remember both the Monkey and the bullet are falling at the same time.
watch that movie
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For more advanced Physics students:
Here is video to convince you. move the video to 45.00 minutes. (utube).
You will watch an experiment that shows the independence of the 2 motions.
video   (you can watch the whole movie of course)

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5) In a projectile motion:
A) Vertically, the acceleration is constant and equal to   __________.
We can use the equations : V =___+___ and d = ___ + _____.
B) Horizontally, the object experiences no ______________ so the velocity is ___________.
V=Vi  and d = ________ .
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Look at the picture again. Along the vertical, the projectile is a free-falling object.
Along the horizontal, the object has a constant velocity. The 2 components of the motion are
completely independent.

HOW LONG ? HOW FAR ? HOW HIGH ? HOW FAST ?

We analyze the horizontal and vertical motions separately, using the big five equations.
With the equations we can answer the questions: How long? How high ? How far ? How fast?
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6) Write the following on index cards:

FIRST, if given, write the initial velocity Vi =  _________  m/s 
Find the components of Vi :     Vix = Vi cos(θ) and Viy= Vi sin(θ) 
θ is the launch angle
see picture above, θ is the angle between the vector Vi (initial velocity) and the positive X-axis.
If the initial velocity is horizontal, θ = 0. If the initial velocity is vertical , θ = 90. 

ALONG THE X-AXIS Horizontal motion : no acceleration (ax = 0) A x-component pointing to the left is negative Remember: Vix = Vi cos(θ) of course if θ = 0 Vix = Vi (the initial velocity is horizontal) if   θ = 90  Vix = 0 (Vi is along the vertical, x =0 )	ALONG THE YAXIS Vertical motion: ay = g = -9.8m/s/s A y-component pointing down is negative. Remember: Viy = Vi sin(θ) of course if θ = 0  Viy = 0  (the initial velocity is horizontal) if   θ = 90   Viy = Vi   (the motion is along the Y-axis only)
Vx = Vix   x-component of the velocity is constant                  = x-component of the initial velocity.	Vy = Viy + a t   or   Vy = Viy - 9.8  t The y-component of the velocity changes  by g= -9.8 ms/s  each second.
x = Vix t    distance along the x-axis. The distance covered every second is the same.	y = Viy t + 0.5 a t2  or  y = Viy t - 4.9 a t2 The distance covered along the y-axis has 2 terms. g= -9.8m/s/s
	Vy2 = Viy2 + 2 a y   or    Vy2 = Viy2 +  19.6 y

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7) Let's answer the question HOW LONG ? (try without hint)
Suppose our footballer (see above image)  throws the ball with an intial velocity Vi = 20m/s@30.
A) find Tmax using the equations. Tmax = time for the ball to reach the top of its parabola.

hint: Focus on the vertical motion of the ball. Viy = 20sin(30)
and at the top Vy = 0.  Use Vy = Viy + a t  with  a = -9.8m/s/s. solve for t.

B) Find the flight time.

hint : flight time = 2 Tmax)

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 8) Let's answer the question: HOW FAR (refer to the image above)? (try without hint)
Find Xmax, the distance covered by the ball, during Tmax, along the horizontal.
given: Vi= 20ms@30 and you have Tmax. (the time for the ball to cover Xmax)

hint: use xmax = Vix t    with Vix = 20cos(30) and t = Tmax.

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9) Let's answer the question: HOW HIGH ?
Find The maximum height reached by the ball H.

hint: At the top Vy = 0m/s. Use Vy² = Viy² + 2 a y  with Viy = 20sin(30), a = -9.8m/s/s . Vy=2 and solve dor y.
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10) Let's answer the question: HOW FAST at t=1.5s ?
To find the vector velocity V at t = 1.5 s you need to find first its components Vx and Vy.
Pythagorean theorem will give you the magnitude of V. Use tangente for the direction.
Trace V. V = _____ @ _____.

Hint: Vx = Vix = 20 cos(30) and Vy = 20sin(30) + a t    with a = -9.8m/s/s and t = 1.5s
V² = Vx² + Vy².  Find V. Trace V using its component to find the quandrant. Find the direction using tan(θ) = Vy/Vx.
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11) optional, for advanced Physics.
Suppose a projectile is thrown at a launch angle of θ and with an initial speed of V.
A) Find the expression of Tmax = _____________ (use only 9.8m/s/s, sin(θ) and V)

B) find the expression of Xmax = ______________ (use sin(θ), cos(θ), 9.8m/s/s and V)

C) In trigonometry it can be shown that 2 sin(θ) cos(θ) = sin (2θ)
Use this formula to write the expression of Xmax = ___________.

D) Do you know that the maximum value for a sinus of cosinus is 1 ? (play with the TI to find out).
Can you find θmax that will maximize the Xmax (the range) ? if θ= θmax the projectile covers its maximum
distance. hint: Use Xmax = V²sin(2θ) / 9.8

E) Find Xmax (using Xmax = V²sin(2θ) / 9.8) for  θ = 20 and θ = 70.
Then for θ = 10 and θ = 80. Then for θ = 30 and θ = 60. Conclusion? (hint:Did you notice the angles sum up to 90)
If you plot sin (2θ) on your TI or by hand you will understand why.
If you don't know how to plot use this applet to draw the function sin (2θ) or come to see me.    
For which angle sin (2θ) reaches its maximum ?
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11) In a projectile motion, a launch angle of 45 degrees will give you the maximum range.
A same range can be reached using 2 different angles as long as they sum up to 90 degrees:




The same range is reached for θ = 50 and θ = ___________________________
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13) A) using the following applet check that if angle= 45 the range is the largest one:
keep the same initial velocity and play with the angle only.
B) Keep the same initial velocity and check that the range is the same
for angles (20 and 70) or( 50 and 40) or (80 and 10) .. as long the angles sum up to 90.
C) Keep the angle the same. Play with the initial velocity. Describe what you observe.
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TRAJECTORY

13) optional , physics major
Can you find y =f(x) ? That will the trajectory of the projectile in a (X,Y) coordinate system ?
Use x = Vix t and y = Viy t - 0.5 g t²
You can also use: tan(θ) = sin (θ) / cos ( θ)

Hint: first from x =  Vix t with Vix = Vcos( θ) solve for t
Then substitute t   in y = Viy t - 0.5 g t²  with Viy = V sin(θ)

The graph y = f(x) is a __________________ because y varies as the square of t
you are stuck :
answers
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14) Try to hit the target.
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LAB FREE-FALL and REACTION TIME

FROM a HEIGHT

An object can be thrown from a height. You still get a projectile motion.
You still need to use the big five and work out the 2 components of the
motion independently.



As you increase the horizontal component of the initial velocity, the range increases.
There is no launch angle. The vertical vertical velocity has an initial value of 0 and
increases by 9.8m/s every second.   Viy = 0m/s  Vy = -9.8 Viy

EXERCISES     as usual try without hints.

15) A) AN object is thrown horizontally with an initial speed of Vix=10 m/s. How far it will drop in 4 seconds?
down is negative. (so a = -9.8m/s/s)
B) take a break and play to have a better comprehension of projectile motion/ (suggested by a student)

16) From a height of 100m, a ball is thrown horizontally with an initial speed of Vix=15 m/s.
How far it travels horizontally in the first 2 seconds.
hint: use x = Vix t

17) A projectile is traveling in a parabolic path for a total of 6s. How does its horizontal velocity 1 s after launch
compare to its horizontal velocity after 4s after launch ?

18) Read over the cartoon. Can help Peter and solve the problem ? so he doesn't get a F ?
hint: First find Vix and Viy (see index card exercice 6), then find Tmax (exercice 7), then the flight time is 2Tmax




19) A ball rolls off a tabletop with an inital velocity of 3m/s (Vix) . If the tabletop is 1.25m (y=1.25m)  above the floor,
a) How long does it take for the ball to hit the floor ?  (Viy = 0, solve for tusing the vertical motion equations)
b) How far does the ball travel horizontally ? (solve for x using t found in a. use the horizontal motion equation with Vix = 3m/s)

20) An object is projected upward with a 30 degrees launch angle and an initial speed of 40 m/s. How long
will it take for the object to reach the top of its trajectory (at the top Vy = 0m/s) ? How high is this (find y = H )?
You did that for 7), 8) and 9) above.
hint: initial horizontal component of the velocity =Vix =  40 cos(30) = 35 m/s
initial vertical component of the velocity = Viy = 40 sin(30) = 20 m/s
consider the horizontal and the vertical motion as being independent from one another.

20) An object is projected with a 30 degrees launch angle and an initial speed of 60 m/s.
How many seconds will it be in the air  ? How far will it ravel horizontally ?  (see 7, 8, 9)
V horizontal initial = Vix  = 52 m/s
V vertical initial = Viy = 30 m/s


21) DRivers is Acapulco dive from a cliff that is 61 m (y)  high.
If the rocks below the cliffs extend outward for 23m (Xmax) what is the minimum
horizontal velocity (Vix) a diver must have to clear the rocks ?
(hint: Viy = 0, Vix= ? , y = 61m, x = 23m - first find the falling time t using the Y motion. )

22) A dart player throws a dart horizontally at a speed of 12.4m/s (Vix)
The dart hits the board 0.32m below the height from which it was thrown.
How far away is the player from the board ?
(hint: Vix = 12.4m/s, y = - 0.32m if you place the player at the origin, use the Y motion to solve for t
then use t and Vix to solve for x  along the X-axis)
 
23) After a bad grade in Physics you throw my Physics book horizontally at 8.0m/s from a cliff
64m high. You calms down as I offer you extra credits. You need your book back.
How far from the base of the cliff should you look for the book of your favorite teacher ?

More challenging :
(optional for advanced Physics)


24) a 3-point jump shot is release 2.2m above the ground, 6.02m from the basket., which is 3.05m high.
A) For launch angles of 30 degrees or 60 degrees (same horizontal displacement), find the speed needed to make
the basket.
B) For which angle is it more important that the player get the speed right?
To explore this question, vary the speed at each angle by 5% and find the change, in the range of the throw.
C) so if the player took Physics 101, is it better an angle of 70degrees or 20 degrees?
(same horizontal displacement)

23) A meteorite is being tracked by radar as it falls through the Earth atmosphere.
When its altitude is 30,000m , the radar screen shows that the meteorite
is traveling with a velocity of 583 m/s at an angle of 28.3 degrees below the horizontal.
How much time elapsed before the meteorite strikes the earth ?
What is the velocity (magnitude and direction) of the meteorite before impact with the earth ?
Use tan(θ) = Vy/Vx


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