91Ó°ÊÓ

When we talk about kinematics in physics, we refer to the study of motion without considering the forces that cause it. In the exercise, a skateboarder moves down a ramp, and her change in speed and position is analyzed. Understanding kinematics helps us calculate parameters like displacement, velocity, and acceleration.

Key aspects of kinematics include:

• Displacement: The change in position from start to finish. Here, it is the length of the ramp (12.0 m).
• Velocity: The speed of an object in a given direction. Our skateboarder reaches a final velocity of 7.70 m/s.
• Acceleration: The rate of change of velocity. It can be constant as in this problem, where the task is to find it from the given data.

The main kinematic equation used here is
\[ v^2 = u^2 + 2as \]where \( v \) is the final velocity, \( u \) the initial velocity, \( a \) the acceleration, and \( s \) the distance traveled.
In this problem, initial velocity \( u \) is 0 since the skateboarder starts from rest. Substituting the given values, we derive the acceleration. This highlights how kinematics can describe the motion precisely using equations.

Acceleration describes how quickly the velocity of an object changes. In the exercise, the skateboarder's acceleration is calculated using the kinematic equation.

Here's a step-by-step breakdown:

• Start with the kinematic equation: \[ v^2 = u^2 + 2as \]
• Substitute the known values: the final velocity \( v = 7.70 \text{ m/s} \), initial velocity \( u = 0 \text{ m/s} \), and distance \( s = 12.0 \text{ m} \).
  
  This gives: \[ 7.70^2 = 0 + 2a(12.0) \]
• Solve for acceleration \( a \) by rearranging the equation: \[ a = \frac{7.70^2}{2 \times 12.0} \approx 2.47 \text{ m/s}^2 \]

This result tells us that, while rolling down the ramp, the skateboarder's speed increases by approximately 2.47 meters per second every second.
Constant acceleration like this makes calculations straightforward, allowing us to use the uniform motion equations of kinematics effectively.

When an object moves down an inclined plane, like a ramp, its motion is affected by the angle of inclination. Here, the ramp is inclined at 25.0° with respect to the ground.

Important considerations in inclined plane analysis:

• The component of motion parallel to the surface determines how the acceleration affects motion along the ramp.
• The angle of inclination, \( \theta \), helps calculate this parallel component of acceleration using trigonometry.

To find the parallel component of acceleration, \( a_{||} \), use the relation:
\[ a_{||} = a \cdot \cos(\theta) \]where \( a = 2.47 \text{ m/s}^2 \) is the acceleration previously calculated, and the angle \( \theta = 25.0° \).
Calculate as follows:\[ a_{||} = 2.47 \cdot \cos(25.0°) \approx 2.24 \text{ m/s}^2 \] This calculation shows how only part of the skateboarder's acceleration acts along the incline's horizontal component, emphasizing the importance of analyzing each component of motion individually.