Question

During each cycle, the velocity \(v\) (in feet per second) of a robotic welding device is given by \(v=2 t-\frac{14}{4+t^{2}}\), where \(t\) is time in seconds. Find the expression for the displacement \(s\) (in feet) as a function of \(t\) if \(s=0\) when \(t=0\).

Short answer

\( s(t) = t^2 - 7 \ln(4 + t^2) + 7 \ln(4) \).

Step-by-step solution

Understand the Given Problem

We have the velocity function of the robotic welding device as \( v = 2t - \frac{14}{4 + t^2} \). We need to find the displacement function \( s(t) \), given that \( s = 0 \) when \( t = 0 \). The displacement \( s(t) \) is the integral of the velocity function with respect to time \( t \).

Set Up the Integral for Displacement

To find the displacement \( s(t) \) as a function of time, we set up the integral as follows: \[ s(t) = \int (2t - \frac{14}{4 + t^2}) \, dt \]. This integral will give us the expression for the displacement.

Integrate Each Term Separately

Now we will integrate each term of the integrand separately: 1. The integral of \( 2t \) is \( \int 2t \, dt = t^2 + C_1 \).2. The integral of \( -\frac{14}{4 + t^2} \) requires a substitution. Let \( u = 4 + t^2 \), hence \( du = 2t \, dt \). The integral becomes \( -7 \int \frac{1}{u} \, du = -7 \ln|u| + C_2 = -7 \ln|4+t^2| + C_2 \).

Combine the Results of Integration

Combine the integrated results to get the general solution for \( s(t) \):\[ s(t) = t^2 - 7 \ln|4 + t^2| + C \], where \( C = C_1 + C_2 \) is the constant of integration.

Apply Initial Condition to Find Constant

Use the initial condition given, \( s(0) = 0 \). Substituting \( t = 0 \) into our general solution, we have:\[ 0 = 0^2 - 7 \ln(4 + 0^2) + C \rightarrow 0 = -7 \ln(4) + C \].Solve for \( C \):\[ C = 7 \ln(4) \].

Write the Final Expression for Displacement

Substitute \( C \) back into the general solution for \( s(t) \):\[ s(t) = t^2 - 7 \ln(4 + t^2) + 7 \ln(4) \].This is the expression for the displacement as a function of time \( t \).

Key concepts

Displacement Function

In physics and calculus, the displacement function describes how an object's position changes over time. It is a crucial concept, especially when analyzing moving objects. Displacement is different from distance; it not only tells us how far an object has moved, but also the direction of the movement. The function is usually derived from a velocity function, through the process of integration.By integrating the velocity function, we can determine how position changes with time. In our example, the robotic device's displacement is derived from its velocity function \[ v = 2t - \frac{14}{4 + t^2} \] This integration results in an expression that gives the location of the device at any point in time. Understanding displacement is fundamental in fields such as mechanics, physics, and engineering for designing and controlling the movement of various entities.

Velocity Integration

Integrating velocity is how we find the displacement of an object. Velocity describes how fast an object is moving and in which direction but doesn't provide the object's exact position. Therefore, we need integration to fill in the gap. This process involves summing up all the small changes in velocity over time, which results in the displacement function.In our given problem, the velocity function is:- \[ v = 2t - \frac{14}{4 + t^2} \] To obtain the displacement function, we perform the integral:- \[ s(t) = \int (2t - \frac{14}{4 + t^2}) \ dt \] This results in a new function that accounts for the total movement over time. The integration process separates and integrates both components of the velocity function. It can involve simple techniques for straightforward terms, like the integral of \( 2t \), or more complex methods for others, requiring substitution for terms like \(-\frac{14}{4 + t^2}\). Each term in the velocity function contributes to the final form of the displacement equation, providing a complete picture of the object's journey.

Initial Condition

The initial condition in an integration problem is a specific value that helps us fine-tune our solution to match physical reality. It establishes a reference point for the displacement. Once we have integrated velocity to find the displacement function, we get a family of curves, all differing by a constant. The initial condition allows us to find this constant.In our exercise, we know that when \( t = 0 \) the displacement \( s = 0 \). This condition means, at time zero, the robotic device started at the origin point. Now, we substitute this into our displacement function to solve for the integration constant \( C \).Applying the initial condition:The displacement function before determining \( C \) is \[ s(t) = t^2 - 7 \ln|4 + t^2| + C \]Plugging in the initial values gives us:\[ 0 = 0 - 7 \ln(4) + C \]Solving yields \[ C = 7 \ln(4) \], which we reinsert into our function:\[ s(t) = t^2 - 7 \ln(4 + t^2) + 7 \ln(4) \]Having this constant ensures our displacement function accurately reflects the specific motion of the robotic device over time.