A 2.90 kg box that is several hundred meters above the surface of the earth is s

Question

A 2.90 kg box that is several hundred meters above the surface of the earth is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope, and this results in a tension in the rope of T(t)=(37.0N/s)t . The box is at rest at t =0. The only forces on the box are the tension in the rope and gravity.

What is the velocity of the box at 1.00 s ?

What is the velocity of the box at 3.00 s ?

What is the maximum distance that the box descends below its initial position?

At what value of t does the box return to its initial position?

Explanation / Answer

here,

from free body diagram

mg - T(t) = ma(t)

a(t) = (mg - T(t)/m

a(t) = g - (37/2.9) t

a(t) = 9.8 - 12.75 t

Velocity as a function of time

a(t) = dV/dt

dV = a(t) dt

V = int of dV = int of (a(t) dt

V = int of ((g - 12.75t)

v(t) = gt - 1275 t^2/2

part A:

V at t= 1s is V(1) = (9.8*1) - (12.75 * 1*1/2)

Vat t = 1 = 3.425 m/s^2

part B :

V at t = 3s,

V(3) = 9.8*3 - (12.75 * 3*3/2)

V(3) = -28 m/s^2

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at t = 0.015 secs it returnss

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