2-Dimensional Motion - Projectiles

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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

2-Dimensional Motion - ProjectilesNow it starts to get more interesting(and dont get freaked out by the equations and subscripts)Projectiles What path do they follow?

http://www.us-inauguration-day-2009.com/human_cannonball.jpgProjectiles follow parabolic paths

Most important thingto remember is that horizontal and verticalmotion are independentof one another.From now on,Horizontal = X directionVertical = Y directionLets look at the horizontal and vertical components individuallyWhich 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? Lets find out

What is the X-component of motion?Same as missing acceleration case for one-dimensional motion.X = V0TBut since we have 2 dimensions, we want to distinguish further between X and Y, soX = V0xTV0 = V naught = same thing as V initialThis is how the book writes it, so I dont want you to get confusedNow lets look at the Y-directionY direction has gravitySo, with no initial vertical speed, the position in the y-direction follows the free fall equation:Y = gt2However, there will be cases where we have an initial vertical speedY = V0yt+ ayt2 = V0yt + gt2 , where g = 9.8m/s2So, lets bring it togetherX stuffY stuff_______________X = horiz positionY = vert positionAx = accel in x-dirAy = accel in y-dirVx = velocity in x-dirVy = velocity in y-dirV0x = Init veloc in x-dirV0y = Init veloc in y-dirVfx = final veloc in x-dirVfy = final veloc in y-dirT = timeT = time

All the 1-D equations you know and love work in 2D!Just use subscripts!When once we hadNow we havev = atvx = axt, vx = v0x + axtx = at2x = axt2 , x = v0x t+ axt2 vf2 = vi2 + 2axvfx2 = vix2 + 2axx

And the same for the Y-directionJust use subscripts!When once we hadNow we havev = atvy = ayt, vy = v0y + ayty = at2y = ayt2 , y = v0y t+ ayt2 vf2 = vi2 + 2ayvfy2 = viy2 + 2ayxAnd 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 motionEven though they are independent, the way in which they work together yields parabolic motionWhen 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+ ayt2 Now, Lets look at some projectiles

http://media.photobucket.com/image/parabolic%20motion/Finer_Kitchens/Marilyn_CakeBalls/scan0008.jpgLets look at the velocity vectors what do you notice?

http://www.phys.ttu.edu/~rirlc/Lecture6.htmlExamine the two different components of the velocity X vs. Y

http://www.phys.ttu.edu/~rirlc/Lecture6.htmlFirst, note the launch angle 0The initial horizontal (X) component of V is given by Vcos()The initial vertical (Y) component of V is given by Vsin()

Examine the two different components of the velocity X vs. Y

http://www.phys.ttu.edu/~rirlc/Lecture6.htmlNow note that the vertical (Y) component of motion changesHorizontal (X) component stays the sameBecause Y component changes, Velocity vector changes both direction and magnitude during flight

Now lets look at some animationsFor motorcycle and archery fun, lets go to

http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html