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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)

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Projectiles – What path do they follow?

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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

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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…

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What is the X-component of motion? Same as ‘missing acceleration’ case for one- dimensional motion. X = V 0 T But since we have 2 dimensions, we want to distinguish further between X and Y, so X = V 0x T “V 0 ” = “V naught” = same thing as “V initial” This is how the book writes it, so I don’t want you to get confused

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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 gt 2 However, there will be cases where we have an initial vertical speed Y = V 0y t+ ½ a y t 2 = V 0y t + ½ gt 2, where g = 9.8m/s 2

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So, let’s bring it together X stuffY stuff_______________ X = horiz positionY = vert position A x = accel in x-dirA y = accel in y-dir V x = velocity in x-dirV y = velocity in y-dir V 0x = Init veloc in x-dirV 0y = Init veloc in y-dir V fx = final veloc in x-dirV fy = final veloc in y-dir T = timeT = time

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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∙tv x = a x t, v x = v 0x + a x t x = ½ at 2 x = ½ a x t 2, x = v 0x t+ ½ a x t 2 v f 2 = v i 2 + 2axv fx 2 = v ix 2 + 2a x x

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And the same for the Y-direction Just use subscripts! When once we had……Now we have v = a∙tv y = a y t, v y = v 0y + a y t y = ½ at 2 y = ½ a y t 2, y = v 0y t+ ½ a y t 2 v f 2 = v i 2 + 2ayv fy 2 = v iy 2 + 2a y x And remember that nine times out of ten, the acceleration in the y-direction (a y ) = g = 9.8m/s 2

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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(t 2 ) x = v 0x t and y = v 0y t+ ½ a y t 2

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Now, Let’s look at some projectiles

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Let’s look at the velocity vectors – what do you notice?

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Examine the two different components of the velocity – X vs. Y First, note the launch angle θ 0 The initial horizontal (X) component of V is given by Vcos(θ) The initial vertical (Y) component of V is given by Vsin(θ)

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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

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Now let’s look at some animations For motorcycle and archery fun, let’s go to… mbattista/proj/projectile.html

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