If someone has a velocity of 32 ft/sec, will they be able to ring the bell( more info below)? At a carnival, a new attraction allows...

The initial velocity ufeet/sec. The bell is at a height of
20 feet above. Whether the contestantwill be able to jump a 20 feet height from the
spring board is the question. So the actual equationis h(t) = ut-(1/2)gt^2 in accordance
with the laws of motion, where h(t) height at time t from the take off, and g is the
acceleration due to gravity.

Given data initial velocity u
= by 32 ft/sec and g = 32 ft/sec^2  a fact
assumed.

Therefore the the equation of motion is h(t) =
utt-(gt^2).

h(t) =
ut-(1/2)t^2.

So if the spring board has a height of 20 ft,
then the model would have to be h(t) =
32t-16t^2+20.

Therefore d(t) = -16t^2-bt+20 and
32t-16t^2+20 must be identical.

So when time t = 0, d(0) =
-16*0^2+32*0+20 = 20 is the platform height which is the initial height of the contest
from where he takes off.

The bell is at 20 ft above the
platform. So the heightof the bell = platform height+20 = 20+20 = 40
ft.

The maximum height d(t) is when d'(t) = 0 and d"(t)
< 0.

d(t) =
ut-1/2gt^2+20 .

d'(t) = u-gt and d"(t) = -g <
0.

d'(t) = 0 gives u-gt= 0. So t = u/g , when
d(t) is maximum. Or d(u/g) is the maximum height the contestant
jumps.

d'(t) = -16*2t+32 and d"(t) = -32
< 0.

When initial velocity u =
32ft/sec:

So d'(t) = -16*2t+32 = 0, or  32t = 32. so t =
1

The maximum height  the contest jumps = d(1) =
-16*1^2+32+20 = 36 < 40ft.

Therefore the contest
does not reach the bell.

If u = 35 ft/sec, then the maximum
height the contestant can jump  = d(u/g) = d(35/32) = -16(35/32)^2+35(35/32)+20 = 39.14
ft < 40 ft. so the contest  falls down before reaching the
bell.

If u = 40, then the contest can jump a maximum height
of d(u/g) = -16(40/32)^2+40(40/32)+20 = 25+20 = 45 ft > 40 ft. So the contest 
reaches the bell  and above a. So he can ring twice the bell while going up and falling
down.

If u = 45 ft, then the maximum height the contestant
reaches is

dt) = d(u/g) = d(45/32) =
-16(45/32^2+45(45/32)+20 = 51.64 ft > 40 ft. So the contest reach a height above
the bell. So he can ring the bell twice. He can ring while going up and falling
off.

If u = 32 ft/sec, then the contest reaches a maximum
height of d(u/g) = d(32/32) = -16*1^2+32+20 = 36 feet = (20+16) ft. So the contest jumps
16 ft above the platform. So if the bell is placed at 10 ft, 12 ft and 15 ft above the
plat form, the contest can reach and ring the bell. But he can not reach the bell place
at the  height of 18 feet as his maximum jump is only 16ft above the
platform.