Question

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=0.2 t ; v(0)=0, s(0)=1$$

Short answer

Answer: The position function of the object is $$s(t) = \frac{1}{3}t^3 + 1$$.

Step-by-step solution

Integrate the acceleration function to find the velocity function

To find the velocity function, integrate the given acceleration function with respect to time: $$v(t) = \int a(t) dt = \int 0.2t dt$$

Calculate the constant of integration for the velocity function

Now apply the initial condition $$v(0) = 0$$ to find the constant of integration (C1) for the velocity function: $$0 = \int 0.2(0)d(0)+ C1$$ So, the constant of integration, $$C1 = 0$$.

Write down the velocity function with the constant

Now we have the velocity function as: $$v(t) = \int 0.2t dt = 0.1t^2 + C1 = 0.1t^2$$

Integrate the velocity function to find the position function

Next, integrate the velocity function with respect to time to find the position function: $$s(t) = \int v(t) dt = \int (0.1t^2) dt$$

Calculate the constant of integration for the position function

Now apply the initial condition $$s(0) = 1$$ to find the constant of integration (C2) for the position function: $$1 = \int 0.1(0)^2 d(0) + C2$$ So, the constant of integration, $$C2 = 1$$.

Write down the position function with the constant

Now we have the position function as: $$s(t) = \int (0.1t^2) dt = \frac{1}{3}t^3 + C2 = \frac{1}{3}t^3 + 1$$ The position function of the object moving along a line is: $$s(t) = \frac{1}{3}t^3 + 1$$

Key concepts

Integrating Acceleration Function

When dealing with motion, understanding the relationship between acceleration, velocity, and position is essential. Acceleration (\(a(t)\)) tells us how the velocity changes over time, but to find the object's velocity (\(v(t)\)) and subsequent position (\(s(t)\)), we need to integrate the acceleration function.

Integration is the process of finding the area under the curve of a function. When we integrate the acceleration function with respect to time, we get the velocity function. Mathematically speaking, if we're given an acceleration function like \(a(t) = 0.2t\), we integrate it to find the velocity: \(v(t) = \frac{1}{5}t^2\). Here, we've essentially summed up all the tiny changes in velocity that occur as time progresses from the start time (often t=0) to any time \(t\).

In the context of motion along a straight line, integrating the acceleration function once gives us the velocity function. However, this is where the 'constant of integration' comes into play, as we'll see in the next section.

Initial Velocity and Position

In most real-world scenarios, we need more than just the general form of the velocity or position function—we need to know exactly where and how fast an object is moving at a specific time. This is where initial conditions come in. Initial velocity (\(v(0)\)) and initial position (\(s(0)\)) provide us with specific values at time \(t = 0\), allowing us to tailor the general solution to a particular situation.

Let's take the initial condition of our earlier example: the velocity function at time zero is given as \(v(0) = 0\). This tells us that the object wasn't moving at the beginning of our observation. Similarly, the initial position \(s(0) = 1\) indicates that the object started one unit of distance (meters, feet, etc.) from the origin of our coordinate system. With these pieces of information, we can solve for the constants of integration which make the functions exact for our problem.

Constants of Integration

The constants of integration are crucial for converting the general solution of an indefinite integral into a specific solution that aligns with the given initial conditions. In the process of integrating an acceleration function like \(a(t) = 0.2t\), we get a velocity function plus an arbitrary constant (\(C_1\)).

Using the initial velocity condition, \(v(0) = 0\), we set the time \(t\) to zero in the velocity function. Through this, we determine that \(C_1\) is zero. A similar process is used to find the second constant (\(C_2\)) when integrating the velocity function to get the position function. Applying the initial position, \(s(0) = 1\), tells us that \(C_2 = 1\). These constants essentially adjust our equations to match the specific starting conditions, anchoring our general solution to a particular event or point in time.

Without these constants, our solutions would remain ambiguous and not applicable to the real-world scenario we're trying to model. By incorporating the constants, we can predict the exact behavior of an object in motion at any given time.