91Ó°ÊÓ

Newton's Laws of Motion are fundamental to understanding how objects move in our world. In this exercise, we primarily draw on Newton's First and Second Laws. These laws help explain why the box moves at a constant speed when a specific force is applied and what happens when that force is removed.

Newton's First Law of Motion, often termed the law of inertia, states that an object at rest stays at rest and an object in motion stays in motion at the same speed and in the same direction unless acted upon by an unbalanced external force. In our scenario, the box continues moving at a constant speed because the worker's applied force balances out the frictional force.

• The box moves at constant speed due to the equal and opposite forces of friction and applied force.
• When the worker stops applying the force, friction becomes the unbalanced force that slows down the box.

Newton's Second Law of Motion gives us the relationship between force, mass, and acceleration, formulated as \( F = ma \). This equation allows us to calculate the force required to maintain constant speed and also the acceleration (or deceleration) once the force is removed.

By applying these laws, we can predict and calculate how the box's motion will change under different conditions.

Kinetic friction is a force that occurs between moving surfaces. It always acts in the direction opposite to the motion. We primarily use the concept of kinetic friction to understand the resisting force that the worker must overcome to keep the box moving.

Kinetic friction can be calculated using the formula \( f_k = \mu_k N \). Here, \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force. The normal force is usually equal to the weight of the object when on a horizontal plane, so \( N = mg \).

• For our box, \( \mu_k = 0.20 \) and the normal force \( N = 109.76 \, \text{N} \).
• This gives a kinetic friction force \( f_k = 21.952 \, \text{N} \).

To keep the box moving at a consistent speed, the worker must exert a force equal to this frictional force, precisely balancing it to prevent acceleration.

Once the force is removed, kinetic friction works to bring the box to a stop by providing a constant deceleration until the motion ceases.

Kinematic equations describe the motion of objects without considering the forces causing the motion. They help us find variables like distance, velocity, and acceleration when other values are known.

In the exercise, we use the kinematic equation \( v^2 = u^2 + 2ad \) to determine how far the box travels after the force stops being applied. This formula relates the final velocity \( v \), initial velocity \( u \), acceleration \( a \), and the distance \( d \).

• Initial velocity \( u \) is 3.50 m/s, and the final velocity \( v \) is 0 m/s because the box comes to a stop.
• The acceleration \( a \) is negative due to deceleration caused by friction, calculated as approximately \(-1.96 \, \text{m/s}^2 \).

By rearranging the equation to solve for \( d \), we find that the box slides approximately 3.12 meters before stopping. This example highlights how kinematic equations serve as powerful tools in predicting motion outcomes in real-world scenarios.