The force applied against an object moving horizontally on a linear path is described by #F(x)=x^2-3x + 3 #. By how much does the object's kinetic energy change as the object moves from # x in [ 0 , 1 ]#?
Newton's second law of motion:
Definitions of acceleration and velocity: Kinetic energy: Answer is:
The second law of motion by Newton:
We get a differential equation by substituting into the existing equation:
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To find the change in kinetic energy of the object as it moves from (x = 0) to (x = 1), we need to calculate the work done by the force (F(x)) over this interval, which will be equal to the change in kinetic energy.
The work done by a force is given by the formula:
[ W = \int_{x_1}^{x_2} F(x) , dx ]
Given that the force (F(x) = x^2 - 3x + 3) and the object moves from (x = 0) to (x = 1), we integrate (F(x)) with respect to (x) over the interval ([0, 1]):
[ W = \int_{0}^{1} (x^2 - 3x + 3) , dx ]
[ W = \left[ \frac{x^3}{3} - \frac{3x^2}{2} + 3x \right]_{0}^{1} ]
[ W = \left( \frac{1^3}{3} - \frac{3(1)^2}{2} + 3(1) \right) - \left( \frac{0^3}{3} - \frac{3(0)^2}{2} + 3(0) \right) ]
[ W = \left( \frac{1}{3} - \frac{3}{2} + 3 \right) - (0 - 0 + 0) ]
[ W = \left( \frac{1}{3} - \frac{3}{2} + 3 \right) ]
[ W = \left( \frac{1}{3} - \frac{9}{6} + \frac{18}{6} \right) ]
[ W = \left( \frac{1}{3} + \frac{9}{6} \right) ]
[ W = \left( \frac{2}{6} + \frac{9}{6} \right) ]
[ W = \frac{11}{6} ]
The change in kinetic energy ((\Delta KE)) is equal to the work done:
[ \Delta KE = W = \frac{11}{6} ]
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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.
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.
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.
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.
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