91Ó°ÊÓ

Gravitational force is a fundamental force of nature that attracts two bodies towards each other. On Earth, it gives weight to physical objects. The force of gravity acting on an object is given by the formula: \[ F = mg \]where \( F \) is the gravitational force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity (approximated as 9.81 m/s² on Earth's surface). In our problem, this gravitational force can be broken into components because the object moves along an inclined plane. Specifically, the component of the gravitational force parallel to the ramp is responsible for accelerating the object down the slope.

Acceleration is the rate at which an object changes its velocity. In the context of our problem, we're interested in how quickly the object increases its speed as it slides down the ramp. The equation for acceleration in this scenario is: \[ a = g \, \sin(\theta) \]where \( a \) is the acceleration, \( g \) is the gravitational acceleration (9.81 m/s²), and \( \theta \) is the angle of the slope (40°). The sine function helps us determine the portion of gravitational force that acts parallel to the ramp, causing acceleration. For our example: \[ a = 9.81 \, \sin(40^\textdegree) \approx 6.30 \text{ m/s}^2 \]

Kinematic equations describe the motion of objects and are derived from Newton's laws of motion. They help us calculate key parameters like displacement, velocity, and time. One critical kinematic equation used in this problem is: \[ v^2 = u^2 + 2as \]where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( s \) is the displacement. Given that the object starts at rest (\( u = 0 \)), we have: \[ v^2 = 0 + 2 \times 6.30 \times 10 \approx 126 \text{ m}^2/\text{s}^2 \]By solving for \( v \), we find: \[ v = \sqrt{126} \approx 11.23 \text{ m/s} \]

An inclined plane is a flat surface tilted at an angle (other than a right angle) to the horizontal. It is a simple machine that allows objects to be moved up or down with less force compared to lifting them vertically. The physics of motion on an inclined plane involves breaking down forces into components parallel and perpendicular to the surface. In this exercise, the inclined plane allows us to explore concepts like component forces, acceleration due to gravity, and kinematic equations. By analyzing these components, we understand that the angle of inclination (\( \theta = 40^\textdegree \)) and the length of the ramp (10 m) significantly impact the object's acceleration and final velocity as it slides down from rest.