91Ó°ÊÓ

Velocity calculation is a crucial step in understanding particle motion. By knowing the acceleration, you can find out how fast a particle moves at a particular location. In this exercise, we're provided with the acceleration function:

• Acceleration, \(a = -(0.1 + \frac{\sin x}{0.8})\).

To find the velocity, use the relationship between acceleration and velocity, which involves integrating the acceleration function with respect to position \(x\). This gives us:

• Velocity, \(v = \int a \, dx \).

Here's how it works:- First, substitute the given acceleration into the integral: - \(v = -\int (0.1 + \frac{\sin x}{0.8}) \, dx \).- Proceed with the integration: - You get an expression of the form: \(v(x) = -0.1x - 0.8\cos x + C\), where \(C\) is your constant of integration.

• This expression tells us how velocity changes with position.
• To determine \(C\), use the initial condition \(v(0) = 1 \mathrm{m/s}\).
• Substitute \(x = 0\) in the integrated velocity expression to solve for \(C\).

After finding \(C\), substitute specific positions into the velocity function to calculate the velocity at those positions.

Integration is a powerful tool in physics for solving problems related to motion. It helps relate different quantities by finding out how one changes in relation to another. In this context, integration allows us to determine velocity from acceleration.

• The process begins with a known acceleration function: - \(a = -(0.1 + \frac{\sin x}{0.8})\).

Integration is performed over the spatial variable \(x\), and it yields the function for velocity:

• The integral gives: - \(\int (0.1 + \frac{\sin x}{0.8}) \, dx\) results in \(-0.1x - 0.8\cos x + C\).

Here’s why integration is crucial:- It helps us transition from a rate of change (acceleration) to an absolute quantity (velocity).- Each integration introduces a constant \(C\), which needs to be determined through initial conditions such as known velocity or position values.

• In this example, the condition \(v(0) = 1\,\mathrm{m/s}\) allowed us to solve for \(C\).

Integrating with respect to position \(x\) is essential when dealing with problems involving non-uniform acceleration, such as one that includes \(\sin x\) in its expression.

Determining the maximum velocity of a particle involves finding when a changing velocity reaches its peak. This happens when the derivative of the velocity with respect to position is zero.

• Given the velocity function: - \(v(x) = -0.1x - 0.8\cos x + 1.8\).

Finding the maximum velocity requires us to find where the velocity's derivative is zero. Differentiate \(v(x)\) with respect to \(x\):

• The derivative, \(\frac{dv}{dx} = -0.1 + 0.8\sin x\).

To find the position \(x\) for maximum velocity, solve the equation:- \(-0.1 + 0.8\sin x = 0\)- Rearranging, \(\sin x = \frac{0.1}{0.8} = \frac{1}{8}\).

• Using inverse trigonometric functions, calculate \(x\).

Finally, substitute this \(x\) back into the velocity function \(v(x)\) to find the numerical value for maximum velocity. This entire process highlights the interplay between differentiation and critical points used widely in physics to understand changing systems.