The force applied against an object moving horizontally on a linear path is described by #F(x)=x^2-x+2 #. By how much does the object's kinetic energy change as the object moves from # x in [ 1, 3 ]#?

Question

The force applied against an object moving horizontally on a linear path is described by #F(x)=x^2-x+2  #. By how much does the object's kinetic energy change as the object moves from # x in [ 1,  3  ]#?

Elena Albarran · Accepted Answer

The work is done is, so it's KE increases by, #26/3# J

#W = int_a^c F(x) dx# #= int_1^3 x^2 - x + 2 dx# #= [ x^3/3 - x^2/2 + 2x ]_1^3 = 26/3#J

Adrian Ayala · Answer

undefined To find the change in kinetic energy, you need to calculate the work done by the force over the given displacement. The work done by a force is given by the integral of the force over the displacement. Then, you can use the work-energy theorem, which states that the work done by the net force on an object is equal to the change in its kinetic energy.

Given:
- Force function: $ F(x) = x^2 - x + 2 $
- Displacement: $ \Delta x = 3 - 1 = 2 $

First, calculate the work done by the force over the displacement:

$$ W = \int_{1}^{3} F(x) \, dx $$

$$ W = \int_{1}^{3} (x^2 - x + 2) \, dx $$

Then, calculate the change in kinetic energy:

$$ \Delta KE = W $$

$$ \Delta KE = \text{Work done by the force} $$

Substitute the values obtained from the integration into the equation for work done to find the change in kinetic energy.