Problem 1 - Applied Calculus

Question

Explain in words what the integral represents and give units. \(\int_{1}^{3} v(t) d t,\) where \(v(t)\) is velocity in meters/sec and \(t\) is time in seconds.

Step-by-Step Solution

Answer

The integral represents the distance traveled in meters from 1 to 3 seconds.

Step 1: Understanding the Integral

The integral \( \int_{1}^{3} v(t) \, dt \) represents the accumulation of velocity \( v(t) \) over time, from \( t = 1 \) second to \( t = 3 \) seconds. In this context, the integral is used to calculate the total distance traveled over the given time interval, as velocity is the rate of change of position.

Step 2: Interpreting Velocity Units

The velocity \( v(t) \) is given in meters per second (m/s). Since the integral of velocity over time gives us the change in position (or distance), the result of the definite integral will be expressed in meters, the unit of distance.

Step 3: Integral and its Units

When integrating a velocity function \( v(t) \) over time, the result yields a distance. Therefore, the units of \( \int_{1}^{3} v(t) \, dt \) are meters. This reflects the total accumulated distance from \( t = 1 \) second to \( t = 3 \) seconds.

Key Concepts

Velocity

Velocity is a measure of how fast something is moving in a specific direction. It is a vector quantity, meaning it has both magnitude and direction. The formula for velocity is usually expressed as:

• Velocity (\( v(t) \)) = Change in position / Change in time

In the context of the given exercise, the velocity function, \( v(t) \), describes how quick and in which direction an object moves at any given time, \( t \).
This particular function is given in meters per second (m/s), a standard unit for measuring speed and direction.
Understanding velocity is crucial because it provides insight into the motion pattern of an object.
In physics and mathematics, knowing the velocity helps compute other important quantities, such as acceleration and distance over time.

Distance

Distance is the total length of the path traveled by an object, regardless of its direction. In physics, calculating the distance an object has traveled over a particular period involves analyzing its velocity.
From the exercise, we compute distance using the integral of the velocity function:

• \( ext{Distance} = \int_{1}^{3} v(t) \, dt \)

This calculation takes the velocity function \( v(t) \) and integrates it over the time interval from \( t = 1 \) second to \( t = 3 \) seconds.
In simpler terms, integrating velocity over time gives you the change in position, which is the distance traveled.
The distance is expressed in meters, based on the given units of velocity, ensuring all measurements are coherent within the problem.
Understanding distance in relation to velocity showcases the interplay between these two concepts, providing a comprehensive picture of motion.

Time Interval

A time interval defines the amount of time over which we observe a particular event or motion. In mathematical terms, it specifies the start and end points over which we are calculating changes, such as change in position due to velocity.

• The given problem defines the time interval from \( t = 1 \) second to \( t = 3 \) seconds.

This tells us that we are interested in studying the object's motion only within this specific period.
Time intervals are important because they allow us to focus on particular segments of motion and analyze changes effectively, such as calculating the total distance covered in that time.
Addressing the concept of time intervals helps understand the boundaries within which the changes occur, reinforcing the application of integrals in real-world situations.
Time intervals, alongside velocity and distance, create a well-rounded model for understanding motion dynamics.