When the heliobacter bacteria swims, it is driven by the rotational motion of its tiny tail. It swims almost at a constant velocity, with small fluctuations due to variations in the rotational motion. As a simple model for the motion, we assume that the bacteria starts with the velocity \(v=10 \mu \mathrm{m} / \mathrm{s}\) at the time \(t=0 \mathrm{~s}\), and is then subject to the acceleration, \(a(t)=a_{0} \sin (2 \pi t / T)\), where \(a_{0}=1 \mu \mathrm{m} / \mathrm{s}^{2}\), and \(T=1 \mathrm{~ms}\). (a) Find the velocity of the bacterium as a function of time. (b) Find the position of the bacterium as a function of time. (c) Find the average velocity of the bacterium after a time \(t=10 T\).

Velocity is a fundamental concept in physics, representing how fast an object moves and in what direction. It is defined as the rate of change of an object’s position with respect to time. For the given problem, our initial step is to derive the velocity function from the acceleration.
We start with the acceleration function given: \[ a(t) = a_0 \sin(2 \pi t / T) \] To find the velocity as a function of time, we integrate the acceleration function. Since the initial velocity, \(v_0 = 10 \mu m/s \), we have: \[ v(t) = \int a(t) \, dt + v_0 \] Performing the integration: \[ v(t) = \int a_0 \ sin(2 \pi t / T) \, dt + v_0 \] The standard integral formula for \(\ sin \) is used here: \[ v(t) = -\frac{a_0 T}{2 \pi} \ cos(2 \pi t / T) + v_0 \] Substituting the given values, we get: \[ v(t) = -\frac{10^{-3}}{2 \pi} \ cos(2000 \ pi t) + 10 \mu m/s \] This is the velocity function, which tells us how the speed of the bacterium changes over time.

Acceleration describes how quickly an object's velocity changes. It involves both the speed and direction of motion. For bacterial motion studied here, the acceleration function is defined as: \[ a(t) = a_0 \ sin(2 \pi t / T) \] Here, \(a_0 \) is 1 \(\mu m/s^2 \), meaning the bacteria's acceleration fluctuates sinusoidally over time with a period of \(T = 1 ms \). Sinusoidal acceleration reflects realistic biological and vibrational movements. This means the tail's motion drives the bacteria in a rhythmic, repetitive pattern. Understanding this helps in predicting the behavior of microorganisms. Integrating to find velocity or position from acceleration is essential in motion mechanics.

Integration is a key mathematical technique in physics, especially for determining quantities like velocity and position from acceleration. When we integrate an acceleration function, we get the velocity function. Similarly, integrating the velocity function gives us the position function.
To illustrate: \[ \text{Starting with acceleration:} \quad a(t) = a_0 \ sin(2 \pi t / T) \] To find velocity (\(v(t) \)), we integrate \( a(t) \): \[ v(t) = \int a(t) \, dt + v_0 \] Getting: \[ v(t) = -\frac{a_0 T}{2 \pi} \ cos(2 \pi t / T) + v_0 \] Next, to find position (\(x(t) \)), we integrate \( v(t) \): \[x(t) = \int v(t) \, dt \] Resulting in: \[ x(t) = -\frac{10^{-3}}{4 \pi^2} \ sin(2000 \pi t) + 10 t \] These integrations help us understand the physical trajectory and speed of an object influenced by a given acceleration.

Average velocity is a measure of the overall change in position over time. It's calculated by dividing the total displacement by the total time taken. For bacterial motion examined here, we seek the average velocity over a specific time interval, \(0 \text{to} \ 10T \. \) The formula for average velocity is: \[ \bar{v} = \frac{1}{10T} \int_0^{10T} v(t) \, dt \] Substituting the velocity function, we get: \[ \bar{v} = \frac{1}{10T} \int_0^{10T} \left( -\frac{10^{-3}}{2 \pi} \ cos(2000 \pi t) + 10 \right) \, dt \] Because the integral of a cosine function over a complete period is zero, only the constant term remains: \[ \bar{v} = \frac{1}{10T} \left[ \left[10t\right]_0^{10T} \right]=\frac{10 \times 10T - 0}{10T} = 10 \mu m/s \] The average velocity tells us the bacterium's consistent overall speed while accounting for fluctuating periods of acceleration.