91Ó°ÊÓ

Particle motion describes how objects move along a path over time. Imagine a particle on a straight line, like our \(\Lambda\) particle moving along the x-axis in this problem.
In this scenario, the particle's position changes continuously over time.
This position is a function of time, denoted as \(x(t)\), meaning it depends on time.The velocity of a particle—as a key element of motion—tells us how fast it changes its position.
In our case, velocity is derived from the given equation \(v^2 = 108 - 9x^2\).
It's the speed of the particle at any point along the x-axis.
Remember that velocity has direction while speed is just the magnitude of velocity.To study more complex motions, these particle motion basics lay the groundwork, enabling us to predict or calculate other aspects like acceleration.

Differentiation in physics is a tool for breaking down how quantities change over time.
It's like analyzing how something evolves as time ticks on.
In our problem, the particle's speed is expressed as a squared function, \(v^2 = 108 - 9x^2\).
To find acceleration, we use differentiation on this speed expression.
Differentiation tells us the rate of change for functions; here, it helps us find how \(v\) changes with time.We employ the chain rule because both velocity \(v\) and position \(x\) are functions of time \(t\):
\(\frac{d}{dt}(v^2) = \frac{d}{dt}(108 - 9x^2)\). Applying the chain rule results in:
\(2v \frac{dv}{dt} = -18x \frac{dx}{dt}\).
Differentiation thus converts our understanding of how fast something happens (velocity) to how much faster or slower that change becomes (acceleration). This capability is crucial in physics for motion analysis.

Calculating acceleration involves determining how a particle's velocity changes over time.
Acceleration in mathematics is presented as the first derivative of velocity with respect to time, \(a(t) = \frac{dv}{dt}\).From our differentiated equation \(2v \frac{dv}{dt} = -18xv\), we solve for acceleration as follows:

• Substitute \(\frac{dx}{dt} = v\)
• This leaves us with \(\frac{dv}{dt} = -9x\)

Thus, the acceleration is expressed as \(a = -9x\, \text{m/s}^2\).
This result informs us that acceleration depends on the position \(x\) along the axis.
Negative acceleration implies the particle slows down or accelerates in the negative direction as \(x\) increases.