2-Dimensional Motion - Projectiles

1. 2-Dimensional Motion - Projectiles Now it starts to get more interesting (and don’t get freaked out by the equations and subscripts)

2. Projectiles – What path do they follow? http://www.us-inauguration-day-2009.com/human_cannonball.jpg

3. 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

4. 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…

5. 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

6. 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

7. 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

8. 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∙tvx = axt, vx = v0x + axt • x = ½ at2 x = ½ axt2 , x = v0xt+ ½ axt2 • vf2 = vi2 + 2ax vfx2 = vix2 + 2axx

9. And the same for the Y-direction • Just use subscripts! • When once we had… …Now we have • v = a∙tvy = ayt, vy = v0y + ayt • y = ½ at2y = ½ ayt2 , y = v0yt+ 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

10. 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 = v0xt and y = v0y t+ ½ ayt2

11. Now, Let’s look at some projectiles http://media.photobucket.com/image/parabolic%20motion/Finer_Kitchens/Marilyn_CakeBalls/scan0008.jpg

12. Let’s look at the velocity vectors – what do you notice? http://www.phys.ttu.edu/~rirlc/Lecture6.html

13. 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

14. 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

15. 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