Projectile Motion Examples

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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
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