Problem 49 - Calculus Single Variable

Question

Suppose a body travels in a line with position \(p(t)\) at time \(t\) and velocity \(v(t)\) at time \(t .\) Show that $$ \int_{a}^{b} v(t) d t=p(b)-p(a) $$

Step-by-Step Solution

Answer

The integral of velocity over \([a, b]\) equals the change in position: \(p(b) - p(a)\).

Step 1: Understanding the relationship between position and velocity

The velocity function, \(v(t)\), represents the rate of change of the position function, \(p(t)\). This relationship implies that the derivative of \(p(t)\) with respect to time, \(t\), is equal to \(v(t)\), i.e., \(\frac{dp(t)}{dt} = v(t)\).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if a function \(F(t)\) is continuous on the interval \([a, b]\) and \(F'(t) = v(t)\), then:\[ \int_{a}^{b} v(t)\, dt = F(b) - F(a) \]. Given \(\frac{dp(t)}{dt} = v(t)\), we can identify \(p(t)\) as the antiderivative of \(v(t)\). Thus, the Fundamental Theorem of Calculus applies, and we can write \[ \int_{a}^{b} v(t)\, dt = p(b) - p(a) \].

Step 3: Conclusion

We have shown that the integral of the velocity function over the interval \([a, b]\) is equal to the change in the position function, i.e., the difference between \(p(b)\) and \(p(a)\). This demonstrates that the area under the velocity-time curve represents the displacement or change in position.

Key Concepts

Position and Velocity Relationship

Understanding the relationship between position and velocity is crucial when studying motion. In physics, the position function, typically denoted as \(p(t)\), describes the location of an object at a certain time \(t\). On the other hand, the velocity function \(v(t)\) reflects how fast the object's position changes over time. To put it simply, velocity is the rate of change of position with respect to time. This is mathematically expressed by saying that the derivative of the position function \(p(t)\) with respect to time \(t\) is equal to the velocity function \(v(t)\).

• Position, \(p(t)\), shows where the object is at time \(t\).
• Velocity, \(v(t)\), shows how quickly the position changes at time \(t\).

This relationship forms the basis for analyzing motion in one dimension.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative results in the original function. In the context of motion, if you know the velocity function, the position function \(p(t)\) can be regarded as an antiderivative of the velocity function \(v(t)\). Here's how it works:

• Given the velocity function \(v(t)\), find a function \(P(t)\) such that \(P'(t) = v(t)\).
• The function \(P(t)\) is then an antiderivative of \(v(t)\).

By finding the antiderivative, we can recover the position function, which helps us analyze how an object has moved over time.

Integration of Velocity

When we integrate the velocity function over a part of its domain, we are effectively summing up all the tiny changes in position to find the total displacement over that interval. According to the Fundamental Theorem of Calculus, if you have a velocity function \(v(t)\) that is continuous over a time interval \([a, b]\), the integral of the velocity function over this interval gives the change in the position function: \[ \int_{a}^{b} v(t)\, dt = p(b) - p(a) \]This equation states that the integral of the velocity function, \(v(t)\), from \(t = a\) to \(t = b\) provides the net change in the position function \(p(t)\) from \(a\) to \(b\).

• The left side of the equation represents the total of all velocity changes over the interval.
• The right side gives the difference in the position \(p(t)\) from time \(a\) to time \(b\).

This integration technique allows us to find how far an object has moved, understanding its displacement.

Displacement Calculation

Displacement is a vector quantity that refers to the change in position of an object. Calculating the displacement involves determining how far out of place an object is; it's the object's overall change in position. When we consider the velocity function over a specific interval and integrate it, we effectively calculate the displacement.To visualize this:

• The area under the velocity-time graph represents the total displacement.
• The difference \(p(b) - p(a)\) shows the total change in position.

Understanding displacement is essential in physics because it provides information about the direction and distance that an object has moved. Displacement can be zero even if the distance traveled is not, which happens if the starting and ending points are the same.