Question

Given the following velocity functions of an object moving along a line, find the position function with the given initial position. $$v(t)=4t+\sin t;s(0)=0$$

Short answer

Question: Determine the position function, s(t), of an object moving along a line with the given velocity function, v(t) = 4t + sin(t), and initial position, s(0) = 0. Answer: The position function of the object is s(t) = 2t^2 - cos(t) + 1.

Step-by-step solution

Integrate the velocity function

Integrate the given velocity function $$v(t)=4t+\sin t$$ with respect to time $$t$$ to obtain the position function $$s(t)$$. Remember to add the constant of integration, which we will call $$C$$: $$s(t)=\int v(t)dt=\int (4t+\sin t)dt=2t^2-\cos t+C$$

Apply the initial condition to find the constant of integration

Now, apply the given initial condition $$s(0)=0$$ to find the value of the constant $$C$$: $$0=2(0{)}^2-\cos (0)+C\Rightarrow C=1$$

Write the final position function

Substitute the value of the constant of integration, $$C=1$$, into the position function obtained in Step 1: $$s(t)=2t^2-\cos t+1$$ This is the position function of the object moving along a line with the given velocity function and initial position.

Key concepts

Velocity Function

When we talk about a velocity function, we are diving into understanding the rate of change of an object's position as it moves over time. In simple terms, the velocity function describes how fast and in what direction an object is traveling at any given moment. For linear motion along a line, this function can be expressed mathematically, like our given function: - $v(t)=4t+\sin t$ Breaking this down: - The term $4t$ represents a linearly increasing part, implying a constant acceleration.- The term $\sin t$ introduces a periodic change, signaling fluctuations in speed akin to waves.Understanding velocity functions is vital as they allow us to derive position functions, which tell us the exact location of our object at any time $t$. This derivation involves integrating the velocity function.

Constant of Integration

Whenever we integrate a function to find another, especially when deriving a position function from a velocity function, we encounter an important concept called the constant of integration. This constant, represented by $C$ in equations, is crucial because it accounts for any initial conditions or offset present in the system. Without it, we could miss crucial information about the function's actual path or position. In our example:- We integrated $v(t)=4t+\sin t$ to find the position function and added $C$ to accommodate unknown starting conditions: $$s(t)=2t^2-\cos t+C$$ - Later, through given initial conditions, we determined that $C=1$. Without the constant $C$, our position function would be incomplete.

Initial Condition

Initial conditions are like set starting points in problems related to differential equations, integration, and motion. They inform us of the specific details at the very beginning, which plays a significant role in shaping the solution. These conditions are crucial in determining values like the constant of integration, ensuring our solutions match reality. In the given problem, the initial condition was:- $s(0)=0$, meaning at time $t=0$, the position was zero.By applying this to our derived function:- We set up the equation $$0=2(0{)}^2-\cos (0)+C$$ which simplified to $C=1$.Thus, this initial condition enabled us to refine and find the exact form of the position function, ensuring accuracy and relevance.