91Ó°ÊÓ

Kinetic energy is a fundamental concept in physics that relates to the motion of an object. It's the energy that an object possesses due to its motion. Essentially, for any object with mass and speed, kinetic energy can be calculated using the formula

• \( KE = \frac{1}{2} m v^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity.

In this specific exercise, we consider the kinetic energy of a particle as it moves from an initial position \( x=0 \) to a final position \( x=l \). Initially, the particle is at rest, so its initial kinetic energy is zero. By the time it reaches \( x=l \), it has acquired a velocity \( v = a\sqrt{l} \). Substituting this velocity into the kinetic energy formula gives us the final kinetic energy as

• \( KE_f = \frac{1}{2} m (a\sqrt{l})^2 = \frac{1}{2} m a^2 l \)

This process helps us understand how the concept of kinetic energy ties into calculating work done in this context.

In physics, velocity is not just about speed; it is speed with a direction. The velocity function describes how the speed and direction of a particle change over time or distance.

In the provided exercise, the velocity function is defined as \( v = a\sqrt{x} \). This equation indicates that the particle's velocity is proportional to the square root of its position \( x \). Here, \( a \) is a constant that scales the velocity based on position.

Understanding this relationship is crucial as it depicts how the particle accelerates over the distance. Initially, at \( x = 0 \), the velocity \( v = a\sqrt{0} \) is zero, meaning the particle starts from rest. As the particle moves and \( x \) increases, the velocity increases according to the square root function, reaching \( v = a\sqrt{l} \) at the position \( x = l \). This information is vital for calculating changes in kinetic energy and thus essential for determining the work done.

Work, in physics, is about how much energy is transferred by a force over a distance. The work-energy theorem is a principle that defines this relationship. It tells us that the work done on an object is equal to the change in its kinetic energy.

In the problem, the initial kinetic energy \( KE_i = 0 \), since the particle is at rest at \( x=0 \). The final kinetic energy \( KE_f \) at \( x=l \) is \( \frac{1}{2} m a^2 l \). By applying the work-energy theorem, we find that the work done \( W \) is given by

• \( W = KE_f - KE_i = \frac{1}{2}m a^2 l - 0 = \frac{1}{2}m a^2 l \)

This calculation illustrates how energy transfers from potential forms—represented by the position-dependent motion—to kinetic energy as the particle moves. Recognizing this change allows us to solve for the total work done by external forces as the particle moves along its path from \( x=0 \) to \( x=l \).