91Ó°ÊÓ

Newton's Second Law of Motion defines acceleration as the rate of change of velocity of an object and can be mathematically expressed by the formula \[ a = \frac{F}{m} \] where:

• \( F \) is the force applied to the object, measured in newtons (N).
• \( m \) is the mass of the object, measured in kilograms (kg).
• \( a \) is the acceleration, measured in meters per second squared (m/s²).

In our problem, the girl pulls the sled with a force of 5.2 N. To find the sled's acceleration, we use its mass of 8.4 kg: \[ a = \frac{5.2\, \text{N}}{8.4\, \text{kg}} \approx 0.619\, \text{m/s}^2 \] Similarly, for the girl, applying the same logic, with her mass being 40 kg, gives us \[ a = \frac{5.2\, \text{N}}{40\, \text{kg}} \approx 0.13\, \text{m/s}^2 \] Acceleration calculations help us understand how different forces affect objects differently based on their mass. Analyzing this principle makes it clear why the sled moves faster than the girl along the ice.

Relative motion is about understanding how the movement of one object is perceived from another moving object. It provides a systematic approach when multiple bodies are in motion towards or away from each other.
In this exercise, both the girl and the sled are moving towards each other. This means we need to calculate their relative acceleration, which is the sum of their individual accelerations, since they are moving in opposite directions towards each other:\[ a_{r} = a_{\text{sled}} + a_{\text{girl}} = 0.619\, \text{m/s}^2 + 0.13\, \text{m/s}^2 = 0.749\, \text{m/s}^2 \]With the relative acceleration known, we can analyze their movement towards the midpoint of their initial separation. The meeting point is determined by how much ground they cover relative to one another, not just how fast each one is moving on its own. Relative motion analysis underscores the importance of considering both objects' movements to find accurate solutions in physics problems.

Kinematic equations are essential in physics to describe motion mathematically. They provide tools for solving various motion scenarios, especially when forces and acceleration are involved.

In this problem, we use a specific kinematic equation to find out when the girl and sled meet:\[ s = \frac{1}{2} a_{r} t^2 \]Here:

• \( s \) is the distance (15 meters between the girl and sled initially).
• \( a_{r} \) is the relative acceleration (0.749 m/s²).
• \( t \) is the time it takes for them to meet.

Rearranging the equation to solve for time \( t \) gives:\[ t^2 = \frac{15 \times 2}{0.749} \approx 40 \]\[ t \approx \sqrt{40} \approx 6.32 \text{ seconds} \]The distance the girl travels towards the sled during this time is calculated as:\[ s_{g} = \frac{1}{2} \times 0.13 \times (6.32)^2 \approx 2.6 \text{ meters} \]Kinematic equations provide the necessary framework to decode how and when objects will meet or separate, which is invaluable for calculating precise motion outcomes.