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How much work would it take to horizontally accelerate an object with a mass of #2"kg"# to #9"m/s"# on a surface with a kinetic friction coefficient of #3#?

Answer 1

It can be anything greater than #81"J"#.

According to the Work-Energy Theorem, the work completed is equal to the object's change in kinetic energy plus the energy lost as a result of friction.

The change in kinetic energy, #"KE"#, is given by
#Delta "KE" = 1/2 (2"kg") [(9"m/s")^2-(0"m/s")^2] = 81"J"#.

It will be "81"J"# no matter how you push the object, so the work done will be at least that.

Conversely, the amount of work done against friction can range from infinitesimally small to infinitely big; let's consider the two extreme cases in an intuitive manner.

The fundamental reason is that the energy lost is the product of the #"kinetic frictional force" = mu m g# (a finite constant) and the distance traveled.
In the case where a huge force is applied, the objects reaches the final speed of #9"m/s"# in almost no time at all. In such a small amount of time, the distance traveled is also approximately zero. Therefore frictional loss is minimal.
On the other end of the spectrum, consider applying a constant force of #mu m g#. Since it is equal in magnitude as the opposing friction force, the net force is zero and the object will not speed up. But as it is moving, energy is constantly being lost by friction. This treatment can be extended for arbitrarily long and you can lose as much energy as you want in this way.
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Answer 2

The work done to horizontally accelerate an object can be calculated using the equation:

[ W = \frac{1}{2} m v^2 ]

where:

  • ( W ) is the work done,
  • ( m ) is the mass of the object (2 kg),
  • ( v ) is the final velocity (9 m/s).

Substituting the given values:

[ W = \frac{1}{2} \times 2 \times (9)^2 ]

[ W = \frac{1}{2} \times 2 \times 81 ]

[ W = 81 ]

Therefore, it would take 81 joules of work to horizontally accelerate the object to 9 m/s.

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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.

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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