91Ó°ÊÓ

An inclined plane is a flat surface that is tilted at an angle to the horizontal. This simple concept can make it easier to move heavy objects up or down because it spreads out the effort over a longer distance. In physics problems, inclined planes are often used to analyze the forces acting on an object.

Let's break down what happens on an inclined plane:

• The gravitational force, which always acts downward, can be split into two components on the slope: one parallel to the slope and one perpendicular.
• The parallel component, calculated as \( mg \sin(\theta) \), is the one causing the object to slide down.
• The perpendicular component doesn’t cause motion along the plane, but it can be used to calculate friction if present.

In our exercise, the block encounters no friction, which is why it smoothly slides up until it stops due to gravity alone. Understanding the forces acting on the inclined plane helps in analyzing the object’s motion.

Motion equations are mathematical formulas that describe how objects move. These equations can predict future behavior if the starting conditions are known. In our exercise, we specifically use the equation: \[v_f = v_i + a t\]where:

• \( v_f \) is the final velocity, which is zero when the object stops.
• \( v_i \) is the initial velocity of 3 m/s as given in the problem.
• \( a \) is the acceleration, which we calculate based on the slope.
• \( t \) is time, which we aim to find.

By rearranging the motion equation and solving for time \( t \), we determine how long it takes for the object to come to a halt on an incline. These equations are powerful tools for understanding the dynamics of motion in any direction, not just on inclined planes.

Acceleration is the rate of change of velocity of an object. In this exercise, the block's acceleration is influenced by gravity acting along the slope. To find the acceleration, we use the equation for the force component along the incline, given by:\[a = -\frac{mg \sin(\theta)}{m}\]

• \( m \) is the mass of the object, in this case, 2.5 kg.
• \( g \) is the acceleration due to gravity, approximately \(9.8 \, m/s^2\).
• \( \theta \) is the angle of the incline, \(45^{\circ}\) in this scenario.

Inserting these values into the equation gives us the actual deceleration (negative acceleration) of approximately \(-6.94 \, m/s^2\), as the motion is against the gravitational pull. Remember, the negative sign indicates that the velocity is decreasing over time. This calculated acceleration, together with the initial velocity, allows us to determine how long the block moves up the slope before it stops.