- #1

Arm

- 16

- 5

- Homework Statement
- Question

An object moves in one dimension according to the function ##x(t) = \frac{1}{3}at^3 ##, where a is a positive constant with units of ##\frac{m}{s^3}.## During this motion, a force F =−bv is exerted on the object, where v is the object’s velocity and b is a constant with units of ##\frac{kg}{s}. ## Which of the following expressions will yield the amount of energy dissipated by this force during the time interval from t=0 and t=T ?

- Relevant Equations
- ##ΔE = \int_{x_1}^{x_2} F(x) dx ##

##F = -b*v##

##x(t) = \frac{1}{3}at^3 ##

##v(t) = at^2##

Since ##F = -b * v## I said ## F(x) = -b*v(x) ##

Also x(0) = 0 and x(T) = ## \frac{1}{3}aT^3 ##

I made the integral ## ΔE = -b* \int\limits_0^\frac{aT^3}{3} v(x) dx ##

I wanted to replace v(x) with v(t) somehow

I first tried ##v(x) = ax^2## but then realized that was wrong

Then I tried using ## x = v*t ## but couldn't do anything with it

Also x(0) = 0 and x(T) = ## \frac{1}{3}aT^3 ##

I made the integral ## ΔE = -b* \int\limits_0^\frac{aT^3}{3} v(x) dx ##

I wanted to replace v(x) with v(t) somehow

I first tried ##v(x) = ax^2## but then realized that was wrong

Then I tried using ## x = v*t ## but couldn't do anything with it