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The force applied against an object moving horizontally on a linear path is described by #F(x)=8-x#. By how much does the object's kinetic energy change as the object moves from # x in [ 2, 4]#?

Answer 1

Benjamin Asiedu

The change in kinetic energy is #=10J#

The change in kinetic energy is equal to the work done,

#DeltaKE=W#

The work done is

#dW=F(x)*dx#

#W=int_2^4(8-x)dx#

#=[8x-x^2/2]_2^4#

#=(32-8)-(16-2)#

#=24-14#

#=10J#

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Answer 2

Nathan Ashcroft

To find the change in kinetic energy, we need to calculate the work done by the force F(x) as the object moves from x = 2 to x = 4. The work done by a force is given by the formula:

[W = \int_{x_1}^{x_2} F(x) , dx]

Given (F(x) = 8 - x), the integral becomes:

[W = \int_{2}^{4} (8 - x) , dx]

[= [8x - \frac{1}{2}x^2]_{2}^{4}]

[= (8(4) - \frac{1}{2}(4)^2) - (8(2) - \frac{1}{2}(2)^2)]

[= (32 - 8) - (16 - 2)]

[= 24 - 14]

[= 10]

Since work done is equal to the change in kinetic energy, the change in kinetic energy is 10 Joules.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.