Your name is Galileo Galilei and you toss a weight upward at 20 feet per second from the top of the Leaning Tower of Pisa (height 184 ft). (a) Neglecting air resistance, find the weight's velocity as a function of time t. v (t) = ft/s (b) Find the height (in feet) of the weight above the ground as a function of time. (c) Where and when will it reach its zenith?

a) v (t) = - 32.2 ft/s² · t + 20 ft/s

b) h (t) = 184 ft + 20 ft/s · t - 16.1 ft/s² · t²

c) The weight will reach its maximum height after 0.62 s. The maximum height will be 190 feet.

Explanation:

Hi there!

a) Since the only force that acts on the weight is the gravity force, the object is under a constant downward acceleration g = - 32.2 ft/s² (it is negative because we consider the upward direction as positive). The acceleration is the variation of the velocity over time (dv/dt). Then:

dv/dt = g

Separating variables:

dv = g dt

Integrating from the initial velocity, v0, to v and from t = 0 to t, we obtain:

v - v0 = g t

v = g t + v0

Then:

v (t) = - 32.2 ft/s² · t + 20 ft/s

b) The velocity of the weight is the variation of the height over time:

dh/dt = v (t)

dh/dt = g t + v0

Separating varibles:

dh = g t dt + v0 dt

Integrating from initial height, h0, to h and from t = 0 to t:

h - h0 = 1/2 · g · t² + v0 · t

h = h0 + v0 · t + 1/2 · g · t²

Then:

h (t) = 184 ft + 20 ft/s · t - 1/2 · 32.2 ft/s² · t²

h (t) = 184 ft + 20 ft/s · t - 16.1 ft/s² · t²

c) When the weight reaches its maximum height, its velocity will be zero. Then, using the equation of velocity we can obtain the time at which the weight is at the maximum height:

v (t) = - 32.2 ft/s² · t + 20 ft/s

0 = - 32.2 ft/s² · t + 20 ft/s

-20 ft/s / - 32.2 ft/s² = t

t = 0.62 s

The weight will reach its maximum height after 0.62 s.

The maximum height will be h (0.62 s):

h (t) = 184 ft + 20 ft/s · t - 16.1 ft/s² · t²

h (0.62 s) = 184 ft + 20 ft/s · (0.62 s) - 16.1 ft/s² · (0.62 s) ²

h (0.62 s) = 190 ft

The maximum height will be 190 feet.

using kinematical equation:

v (t) = v0+gt

where v0 is initial velocity and g is the acceleration

we know v0=20fts⁻¹

g=-32fts⁻²

putting values

v (t) = 20-32t

b) now 2nd kinematical equation states that:

s (t) = s0+v0t+1/2gt²

where s0 is initial height which is 184ft

s (t) = 184+20t+1/2 (-32) t²

=84+20t-16t²

now for calculating the time it reaches to zenith when v (t) = 0

so,

20-32 (t) = 0

by simplifying:

t=0.625sec

s (0.625sec) = 184+8 (0.625) - 16 (0.625) ²

s (0.625) = 182.75ft