Download 2-Dimensional Motion

2-Dimensional Motion Projectiles
Now it starts to get more interesting
(and don’t get freaked out by the equations and subscripts)
Projectiles – What path do they
follow?
http://www.us-inauguration-day-2009.com/human_cannonball.jpg
Projectiles follow parabolic paths
Most important thing
to remember is that
horizontal and vertical
motion are independent
of one another.
From now on,
Horizontal = X direction
Vertical = Y direction
Let’s look at the horizontal and vertical
components individually
• Which way does gravity point? DOWN!!!
• So, there is no gravity in the horizontal
direction (x-direction)
• There is only gravity in the vertical direction
(y-direction)
• So, in general, there is no acceleration in the
horizontal direction (x-direction)
• Take a moment to let that sink in.
• This is where parabolic motion comes from.
Why? Let’s find out…
What is the X-component of motion?
• Same as ‘missing acceleration’ case for onedimensional motion.
• X = V0T
• But since we have 2 dimensions, we want to
distinguish further between X and Y, so
• X = V0xT
• “V0” = “V naught” = same thing as “V initial”
• This is how the book writes it, so I don’t want you
to get confused
Now let’s look at the Y-direction
• Y direction has gravity
• So, with no initial vertical speed, the position
in the y-direction follows the free fall
equation:
• Y = ½ gt2
• However, there will be cases where we have
an initial vertical speed
• Y = V0yt+ ½ ayt2 = V0yt + ½ gt2 , where g = 9.8m/s2
So, let’s bring it together
•
•
•
•
•
•
•
X stuff
X = horiz position
Ax = accel in x-dir
Vx = velocity in x-dir
V0x = Init veloc in x-dir
Vfx = final veloc in x-dir
T = time
Y stuff_______________
Y = vert position
Ay = accel in y-dir
Vy = velocity in y-dir
V0y = Init veloc in y-dir
Vfy = final veloc in y-dir
T = time
All the 1-D equations you know and
love work in 2–D!
•
•
•
•
• Just use subscripts!
When once we had…
…Now we have
v = a∙t
vx = axt, vx = v0x + axt
x = ½ at2
x = ½ axt2 , x = v0x t+ ½ axt2
vf2 = vi2 + 2ax
vfx2 = vix2 + 2axx
And the same for the Y-direction
•
•
•
•
•
• Just use subscripts!
When once we had…
…Now we have
v = a∙t
vy = ayt, vy = v0y + ayt
y = ½ at2
y = ½ ayt2 , y = v0y t+ ½ ayt2
vf2 = vi2 + 2ay
vfy2 = viy2 + 2ayx
And remember that nine times out of ten, the
acceleration in the y-direction (ay) =
g = 9.8m/s2
So then why is projectile motion
parabolic?
• Because of the interaction between X and Y
components of motion
• Even though they are independent, the way in
which they work together yields parabolic motion
• When there is acceleration in the y-direction
(gravity) and NO acceleration in the x-direction,
you have equation of the form x = f(t) and y = f(t2)
• x = v0x t and y = v0y t+ ½ ayt2
Now, Let’s look at some projectiles
http://media.photobucket.com/image/parabolic%20motion/Finer_Kitchens/Marilyn_CakeBalls/scan0008.jpg
Let’s look at the velocity vectors –
what do you notice?
http://www.phys.ttu.edu/~rirlc/Lecture6.html
Examine the two different components
of the velocity – X vs. Y
•First, note the launch
angle θ0
•The initial horizontal (X)
component of V is given by
Vcos(θ)
•The initial vertical (Y)
component of V is given by
Vsin(θ)
http://www.phys.ttu.edu/~rirlc/Lecture6.html
Examine the two different components
of the velocity – X vs. Y
•Now note that the vertical (Y)
component of motion changes
•Horizontal (X) component
stays the same
•Because Y component
changes, Velocity vector
changes both direction and
magnitude during flight
http://www.phys.ttu.edu/~rirlc/Lecture6.html
Now let’s look at some animations
• For motorcycle and archery fun, let’s go to…
• http://www.mhhe.com/physsci/physical/gia
mbattista/proj/projectile.html