91Ó°ÊÓ

Newton's Second Law is a fundamental principle that explains how the motion of an object changes when it is subjected to forces. This law states the relationship between an object's mass, its acceleration, and the force applied. Mathematically, it is represented as:

• \( F = ma \)

Here, \( F \) is the force applied to an object, \( m \) is the object's mass, and \( a \) is the acceleration produced.
In the given exercise, we observe this law in action when you pull your friend across the ice. Your friend has a mass of 65.0 kg and you apply a force that results in an acceleration of 2.00 m/s². By using this law, the force you applied is calculated:

• \( F = 65.0 \text{ kg} \times 2.00 \text{ m/s}^2 = 130.0 \text{ N} \)

This force is what helps your friend move from a state of rest to a speed of 6.00 m/s.

The concepts of work and energy help us understand how forces can cause an object to move and gain speed. Work is defined as the process of a force causing an object to move in the direction of the force. It is calculated as:

• \( W = F \, d \)

where \( W \) is the work done, \( F \) is the force applied, and \( d \) is the displacement of the object.
In this scenario, the work done by you on your friend is moving her across the ice. Given that you applied a force of 130.0 N and she moved a distance of 9.00 m, the work done is:

• \( W = 130.0 \text{ N} \times 9.00 \text{ m} = 1170.0 \text{ J} \)

The unit "Joule" is a measure of energy, so 1170.0 J of energy was used to accelerate your friend.

Kinematic equations describe the motion of objects and relate the various aspects of motion: displacement, speed, velocity, and time. In this case, these equations help determine the acceleration and displacement of your friend.
To find acceleration, we use:

• \( a = \frac{v_f - v_i}{t} \)

where \( v_f \) is the final velocity (6.00 m/s), \( v_i \) is the initial velocity (0 m/s), and \( t \) is the time (3.00 s).

• \( a = \frac{6.00 \text{ m/s} - 0 \text{ m/s}}{3.00 \text{ s}} = 2.00 \text{ m/s}^2 \)

To calculate the displacement, we use:

• \( d = \frac{1}{2} a t^2 \)

Substituting the known values:

• \( d = \frac{1}{2} \times 2.00 \text{ m/s}^2 \times (3.00 \text{ s})^2 = 9.00 \text{ m} \)

These equations help us understand how the applied force results in your friend's movement, connecting force with motion through the use of kinematic principles.