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Projectile Motion Examples Conceptual Example 3-6 v0x • Demonstration!! Conceptual Ex. 3-7: Wrong Strategy • “Shooting the Monkey”!! • Video Clip!! Example 3-8 • Range (R) of projectile Maximum horizontal distance before returning to ground. Derive a formula for R. • Range R the x where y = 0! • Use vx = vx0 , x = vx0 t , vy = vy0 - gt y = vy0 t – (½)g t2, (vy) 2 = (vy0)2 - 2gy • First, find the time t when y = 0 0 = vy0 t - (½)g t2 t = 0 (of course!) and t = (2vy0)/g • Put this t in the x formula: x = vx0 (2vy0)/g R R = 2(vx0vy0)/g, vx0= v0cos(θ0), vy0= v0sin(θ0) R = (v0)2 [2 sin(θ0)cos(θ0)]/g R = (v0)2 sin(2θ0)/g (by a trig identity) Example 3-9, A punt! • v0 = 20 m/s, θ0 = 37º • vx0= v0cos(θ0) = 16 m/s, vy0= v0sin(θ0) = 12 m/s Proof that projectile path is a parabola • x = vx0 t , y = vy0 t – (½)g t2 Note: The same time t enters both equations! Eliminate t to get y as a function of x. Solve x equation for t: t = x/vx0 Get: y = vy0 (x/vx0) – (½)g (x/vx0)2 Or: y = (vy0 /vx0)x - [(½)g/(vx0)2]x2 Of the form y = Ax – Bx2 A parabola in the x-y plane!! Example: Rescue helicopter drops supplies A rescue helicopter wants to drop a package of supplies to isolated mountain climbers on a rocky ridge 200 m below. If the helicopter is traveling horizontally with a speed of 70 m/s (250 km/h), a) How far in advance of the recipients (horizontal distance) must the package be dropped? b) Instead, someone in the helicopter throws the package a horizontal distance of 400 m in advance of the mountain climbers. What vertical velocity should the package be given (up or down) so that it arrives precisely at the climbers’ position? c) With what speed does the package land in the latter case? Problem 31 Example: That’s Quite an Arm! Problem: A stone is thrown from the top of a building at an angle θ0 = 26° to the horizontal and with an initial speed v0 = 17.9 m/s, as in the figure. The height of the building is 45.0 m. a) How long is the stone "in flight"? b) What is the speed of the stone just before it strikes the ground? Example: Stranded Explorers Problem: An Alaskan rescue plane drops a package of emergency rations to a stranded party of explorers, as shown in the picture. If the plane is traveling horizontally at v0 = 42.0 m/s at a height h = 106 m above the ground, where does the package strike the ground relative to the point at which it is released? v0 = 42 m/s h