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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. - PowerPoint PPT Presentation
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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 thingto remember is that horizontal and verticalmotion are independentof one another.
From now on,Horizontal = X directionVertical = 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 one-dimensional 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 Y stuff_______________
• X = horiz position Y = vert position• Ax = accel in x-dir Ay = accel in y-dir
• Vx = velocity in x-dir Vy = velocity in y-dir
• V0x = Init veloc in x-dir V0y = Init veloc in y-dir
• Vfx = final veloc in x-dir Vfy = final veloc in y-dir• T = time 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 = vi
2 + 2ax vfx2 = vix
2 + 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 = vi
2 + 2ay vfy2 = viy
2 + 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
http://www.phys.ttu.edu/~rirlc/Lecture6.html
•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)
Examine the two different components of the velocity – X vs. Y
http://www.phys.ttu.edu/~rirlc/Lecture6.html
•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
Now let’s look at some animations
• For motorcycle and archery fun, let’s go to…
• http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html