Question

Given the following velocity functions of an object moving along a line, find the position function with the given initial position. $$v(t)=2 t+4 ; s(0)=0$$

Short answer

Answer: The position function is $$s(t) = t^2 + 4t$$.

Step-by-step solution

Integrate the velocity function

To find the position function, we need to integrate the velocity function. We know that the position function is the integral of the velocity function. So we need to determine the integral of $$v(t)=2t+4$$. \(\displaystyle \int{v(t) \, dt} = \int{(2t+4) \, dt}\)

Perform the integration

Now, let's integrate the function: \(\displaystyle \int{(2t+4) \, dt} = 2\int{t \, dt} + 4\int{1 \, dt}\) Using the power rule for integration, we have: \(s(t) = t^2 + 4t + C\) where C is the constant of integration.

Use the initial position to find the constant of integration

We are given the initial position of the object to be $$s(0)=0$$. Let's use this information to find the constant of integration, C: \(s(0) = 0^2 + 4(0) + C = 0\) Solving for C, we find that: \(C = 0\)

Write the position function

Now that we have found the constant of integration, we can write the position function. Since C = 0, we can ignore it in the position function: \(s(t) = t^2 + 4t\) This is the final position function for the given velocity function and initial position.

Key concepts

Understanding Integration

Integration is a core concept in calculus that helps us find the total accumulation of quantities.
It is essentially the reverse process of differentiation, where we look for a function whose derivative returns us to our original function.
When dealing with motion, integration is incredibly useful.
If we have a velocity function, integrating it allows us to discover the position function of an object.

• The integral is represented as a stretched 'S' symbol: \( \displaystyle \int \)
• The function to be integrated is followed by \( dt \) (or appropriate variable), indicating integration with respect to time.
• The constant of integration, commonly represented as 'C,' accounts for any initial conditions or constants present in the original function.

Applying integration to the velocity function \( v(t) = 2t + 4 \), we find the position function, simply by solving \ \displaystyle \int (2t+4) \, dt. \"

Diving into the Velocity Function

A velocity function describes how an object's speed changes over time.
It is a central concept in motion analysis, enabling predictions about future positions.
The velocity function helps us understand both the speed and direction of a moving object.

• Velocity is a vector quantity, meaning it includes both magnitude and direction.
• For our exercise, the given velocity function is \( v(t) = 2t + 4 \), which is continuous and linear.
• By integrating the velocity function, we can determine the total distance covered or change in position over a given time period.

The velocity function changes over time; hence, integration allows us to smoothly capture its impact on position.

Connecting to the Position Function

The position function provides detailed insights into the original location from a given point over time.
Once we have the position function, determining the location of an object at any point in time becomes easy.
It essentially represents the antiderivative of the velocity function.

• The initial position is crucial, as it gives us a starting 'point' for the position function.
• In this example, we have the initial position \( s(0) = 0 \), allowing us to find the constant of integration.
• By integrating the given \( v(t) \), and using the initial condition, we derive \( s(t) = t^2 + 4t \) as the position function.

This position function gives a complete description of where the object will be along the line at any given time 't'.
This linking of the velocity function through integration to the position function embodies the beauty of calculus.