How Do You Calculate Velocity in Calculus-Based Physics?

[student34]

Homework Statement

A 3kg box is held up high in the air by a a rope with negligible mass. The rope's tension is depends on the function of time T(t) = (36N/s)*t. The box is at rest at t = 0. Only the forces tension and gravity act on this box.

a) What is the velocity at t = 1.0s?
b) What is the velocity at t = 3.0s?

Homework Equations

I think that my acceleration formula is the problem, but I don't know why.

a(t) = (2.2m/s^3)t - 9.8m/s^2 I took out the mass to get the box's acceleration and then I just broke up the applied acceleration from the constant gravitational acceleration.

v(t) = (1.1m/s^3)t^2 - (9.8m/s^2)t This is my integral of the acceleration.

The Attempt at a Solution

a) a(1s) = 1.1m/s - 9.8m/s = -8.7m/s but the book's answer is -3.8m/s.

I just can't grasp how I am wrong.

 

[voko]

How did you get 2.2 m/s^3 from 36 N/s?

 

[student34]

How did you get 2.2 m/s^3 from 36 N/s?

ƩFy = T - mg = ma; a = 12m/s^2 - 9.8m/s^2 = 2.2m/s^2, so I divided out the mass to get the box's acceleration of 12m/s^2. Then, I ...

Lol, ok, in my attempt to answer your question I see how it can work I can get the right answer. Thanks for asking :)

 

[voko]

You are welcome :)

 

[Nickie]

Your approach to finding the acceleration is correct, but your calculation for the velocity at t=1.0s is incorrect. The correct calculation should be:

v(1s) = (1.1m/s^3)(1s)^2 - (9.8m/s^2)(1s) = -8.7m/s

The book's answer of -3.8m/s is incorrect. The correct answer is -8.7m/s.

For part b), the velocity at t=3.0s can be calculated using the same formula:

v(3s) = (1.1m/s^3)(3s)^2 - (9.8m/s^2)(3s) = 19.6m/s

This means that at t=3.0s, the box will be moving downwards with a velocity of 19.6m/s.

It is important to note that the velocity and acceleration calculations will change if the mass of the box is taken into account. In this case, the acceleration formula would be a(t) = (36N/s)/m - 9.8m/s^2 and the velocity formula would be v(t) = (36N/s)t/m - (9.8m/s^2)t.