A box is given a push so that it slides across the floor. How far will it go, given that the coefficient of kinetic friction is 0.24 and the push imparts an initial speed of 4.2 m/s?

It will travel a distance X such that:

Work done against friction = Initial kinetic energy

Uk*M*g*X = (1/2)MVo^2
X = Vo^2/(2*Uk*g*)

Uk is the kinetic friction coefficient.

Note that the mass M cancels out.

To determine how far the box will go, we need to first calculate the deceleration due to friction acting on the box. Once we have the deceleration value, we can then use it to determine the distance traveled by the box using the equation of motion.

Here's how to calculate the deceleration due to friction:

1. Use the formula for the force of friction:
Frictional Force = Coefficient of Kinetic Friction × Normal Force

The normal force is the force exerted by the surface on the box, which is equal to the weight of the box in this case (assuming the box is on a horizontal surface).

2. Calculate the weight of the box:
Weight = Mass × Gravity

Assuming the mass of the box is known, multiply it by the acceleration due to gravity (approximated as 9.8 m/s²).

3. Substitute the weight of the box into the formula for the force of friction and solve for the frictional force.

4. Next, use Newton's second law of motion:
Frictional Force = Mass × Acceleration

Rearrange the equation to solve for acceleration.

5. Substitute the frictional force and mass of the box into the equation and solve for the acceleration due to friction.

Now that we have the deceleration due to friction, we can calculate the distance traveled by the box:

1. Use the equation of motion:
Distance = (Initial Velocity² - Final Velocity²) / (2 × Acceleration)

Since the box comes to a stop, the final velocity is zero.

2. Substitute the values into the equation and solve for the distance.

Following these steps will give you the distance the box will travel after it is given a push and starts sliding across the floor.