The force applied against an object moving horizontally on a linear path is described by #F(x)=4e^x+ 1/x #. 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)=4e^x+  1/x #. By how much does the object's kinetic energy change as the object moves from # x in [ 1,  3 ]#?

Charlotte Aaron · Accepted Answer

The change in kinetic energy is #=70.57Jf#

We need

#inte^xdx=e^x+C#

#int(1/x)dx=ln(|x|)+C#

The change in Kinetic Energy is equal to the work done.

#DeltaE=W=F*d#

#W=int_1^3(4e^x+1/x)dx#

#=[4e^x+lnx]_1^3#

#=(4e^3+ln3)-(4e+ln1)#

#=70.57J#

Axel Acevedo · Answer

undefined To find the change in kinetic energy, integrate the force function over the given interval. The change in kinetic energy is the difference between the kinetic energy at the upper and lower limits of the interval.

$$ \Delta KE = \int_{1}^{3} F(x) \,dx = \int_{1}^{3} (4e^x + \frac{1}{x}) \,dx $$

Julia Aguilar · Answer

undefined To find the change in the object's kinetic energy as it moves from x = 1 to x = 3, you need to evaluate the work done by the force over that interval. The work done by the force is equal to the change in kinetic energy.

The formula for work done by a force is given by the definite integral of the force function over the interval [1, 3].

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

Given the force function $ F(x) = 4e^x + \frac{1}{x} $, integrate it over the interval [1, 3] to find the work done:

$$ W = \int_{1}^{3} (4e^x + \frac{1}{x}) \, dx $$

Evaluate this integral to find the work done.

Once you have the work done, you can equate it to the change in kinetic energy of the object. The change in kinetic energy is given by the difference in kinetic energy at the endpoints of the interval:

$$ \Delta KE = KE(3) - KE(1) $$

Where $ KE(x) $ is the kinetic energy function.

So, to find the change in kinetic energy, subtract the kinetic energy at $ x = 1 $ from the kinetic energy at $ x = 3 $.