Miranda throws a set of car keys to her brother who is standing a third story high with his hands 38 feet above thr ground. If Miranda throws the keys with an initial upword velocity of h=-16t^2+40t+5 gives the height h of the keys after t seconds
how long does it take the keys to reach their highest height?
how high do the keys reach?
will her brother be able to catch the keys?
Not having heard anything further from you, I offer the following with some editing/clarification making this a bit more understandable.
The upward velocity, though not given directly, is implied in the expression you give.
Miranda throws a set of car keys upward to her brother who is standing in a third story apartment with his outstretched hands 38 feet (three stories-?) above the ground. The keys leave her hand 5 feet (implied in the expression you gave for “h”) above the ground, the launch point, with an upward velocity Vo = 40 feet per second (implied in the expression you gave for “h”.)
The height "h" reached by the upward propelled keys at any time “t” seconds derives from the expression h = Vo(t) - gt^2/2 = Vo(t) - 16t^2 where h = the attained height in feet from the launch point, Vo = the initial velocity in feet per second, g = the acceleration due to gravity, 32 ft/sec^2 and t = the time in seconds.
You give the expression for h = 5 - 16t^2 + 40t which includes the above ground launch height of 5 feet and the initial velocity of 40 feet per second.
The following equations of uniform motion apply to rising and falling
For rising bodys,
Vf = Vo - gt
s = Vot - gt^2/2
Vf^2 = Vo^2 - 2gs.
For falling bodys,
Vf = Vo + gt
s = Vot + gt^2/2
Vf^2 = Vo^2 + 2gs
How long does it take the keys to reach their highest height?
From Vf = Vo – gt, 0 = 40 – 32t making the rise time to maximum height t = 1.25 seconds.
How high do the keys reach?
From h = Vo t – g(t^2)/2, h = 40t – 16t^2 + 5 (launch height) = 30 feet, short of the outstretched hands of her brother.
Will her brother be able to catch the keys? NO
More interesting:
What upward velocity will enable her brother to catch the keys?
From Vf^2 = Vo^2 – 2gh = 0 = Vo^2 – 2(32)33 making Vo = 45.95 feet per second or 46 fps for short.
Then, from Vf = Vo – gt, 0 = 46 – 32t making t = 1.4375 seconds to maximum height.
The height reached h = 46(1.4375) – 16(1.4375)^2 + 5 = 66.125 – 33.06 + 5 = 38.06 feet. KEYS SAVED - with ¾ of an inch to spare.
I truly hope this has been of some help to your understanding problems of this sort.
