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 0.17 and the push imparts an initial speed of 4.1 m/s?

To determine how far the box will go, we need to consider the forces acting on it and the relationship between force, friction, acceleration, and distance.

First, let's list the forces acting on the box:
1. The force of the push: This force imparts an initial speed to the box.
2. The force of kinetic friction: This force opposes the motion of the box and eventually brings it to a stop.

The force of kinetic friction can be calculated using the equation:
Friction = coefficient of friction * normal force

The normal force is the force exerted by the floor on the box to support its weight. Since the box is sliding on a horizontal surface, the normal force is equal to the weight of the box.

Next, we'll consider the relationship between force, friction, and acceleration:
The net force acting on the box is equal to the force of the push minus the force of friction. According to Newton's second law, force equals mass times acceleration (F = m * a).

Since the force of friction opposes motion, it has the opposite direction of the push, so we subtract it: Net force = force of push - force of friction.

Finally, we know that acceleration is the rate of change of velocity. Since the box eventually comes to a stop, the final velocity is zero. We can calculate the deceleration using the equation:
Acceleration = change in velocity / time.

Now, let's put everything together to find the distance the box will travel:

1. Calculate the force of friction:
Friction = coefficient of friction * normal force

2. Calculate the net force:
Net force = force of push - force of friction

3. Calculate the deceleration:
Acceleration = change in velocity / time

4. Use the kinematic equation to find the time needed to reach zero velocity:
0 = initial velocity - (acceleration * time)

5. Solve the equation for time.

6. Calculate the distance traveled:
Distance = (initial velocity * time) - (0.5 * acceleration * time^2)

By following these steps, you should be able to determine the distance the box will travel given the coefficient of kinetic friction and the initial speed.

work done by friction=initial KE

forcefriciton*distance= 1/2 m v^2
mu*mg*distance= 1/2 m 4.1^2
solve for distance.