# How much work would it take to horizontally accelerate an object with a mass of #2# #kg# to #4# #ms^-1# on a surface with a kinetic friction coefficient of #1#?

The work done is

The work done is equal to the kinetic energy

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

[ 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 (4 m/s).

The work done against friction can be calculated using the formula:

[ W_f = f \cdot d ]

where:

- ( W_f ) is the work done against friction,
- ( f ) is the force of friction,
- ( d ) is the distance over which the force is applied.

The force of friction can be calculated using:

[ f = \mu \cdot N ]

where:

- ( \mu ) is the coefficient of kinetic friction (1),
- ( N ) is the normal force.

The normal force can be calculated as ( N = m \cdot g ), where ( g ) is the acceleration due to gravity (9.8 m/s²).

Substituting the given values:

[ N = (2 kg) \cdot (9.8 m/s^2) = 19.6 N ]

[ f = (1) \cdot (19.6 N) = 19.6 N ]

[ W_f = (19.6 N) \cdot (d) ]

Now, we need to solve for ( d ) using the equation for kinetic energy:

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

[ W = \frac{1}{2} (2 kg) (4 m/s)^2 ]

[ W = 16 J ]

[ W = W_f ]

[ 16 J = (19.6 N) \cdot (d) ]

[ d = \frac{16 J}{19.6 N} ]

[ d ≈ 0.816 m ]

So, it would take approximately 0.816 meters of work to horizontally accelerate the object to 4 m/s on a surface with a kinetic friction coefficient of 1.

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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.

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