# How much work would it take to horizontally accelerate an object with a mass of #4 kg# to #2 m/s# on a surface with a kinetic friction coefficient of #3 #?

To do this, we'll use the equation

where

The kinetic friction force is thus

The magnitude of the acceleration is

It starts from rest, so we have

Using

we have

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

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

where:

- ( W ) is the work done,
- ( m ) is the mass of the object (4 kg),
- ( v_f ) is the final velocity (2 m/s).

First, calculate the initial velocity using the friction force:

[ F_f = \mu_k \cdot N ]

[ a = \frac{F_f}{m} ]

[ v_i = \sqrt{2a \cdot d} ]

[ W = \frac{1}{2} m (v_f^2 - v_i^2) ]

where:

- ( F_f ) is the frictional force,
- ( \mu_k ) is the coefficient of kinetic friction (3),
- ( N ) is the normal force,
- ( a ) is the acceleration,
- ( v_i ) is the initial velocity,
- ( d ) is the distance over which the frictional force acts.

Given that the object starts from rest and there is no mention of the distance over which the frictional force acts, the work done cannot be accurately determined without that information.

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