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