Problem 74 - Calculus with Concepts in Calculus

Question

Show that if a particle has zero velocity for any given period of time, then it stands still during that period.

Step-by-Step Solution

Answer

If a particle's velocity is zero, its position does not change, so it stands still.

Step 1: Define Velocity

Velocity is the rate of change of the position of a particle with respect to time. It can be expressed mathematically as \( v(t) = \frac{d}{dt}(x(t)) \), where \( x(t) \) is the position of the particle at time \( t \).

Step 2: Zero Velocity Condition

If the particle has zero velocity, then \( v(t) = 0 \). This implies that the rate of change of position with respect to time is zero over the given period.

Step 3: Interpret Zero Rate of Change

A zero rate of change in position means that the position function \( x(t) \) is constant over the specified period. This is because the derivative of a constant is zero.

Step 4: Conclusion from the Interpretation

Since the position \( x(t) \) does not change while the velocity is zero, the particle does not move. Therefore, the particle stands still during the period where its velocity is zero.

Key Concepts

Velocity

Velocity is an essential concept in calculus, commonly used to describe how fast something is moving and in which direction. Specifically, it measures the rate at which the position of an object changes over time. In mathematical terms, velocity is defined as the derivative of the position function with respect to time. This can be expressed as follows:

• Let the position of a particle at time \( t \) be represented by \( x(t) \).
• The velocity \( v(t) \) is then given by \( v(t) = \frac{d}{dt}(x(t)) \).

This formula tells us that velocity considers not just how far an object moves, but also how quickly the movement occurs. Whenever you see a derivative like \( \frac{d}{dt}(x(t)) \), you’re looking at a rate of change—how one variable changes with respect to another over time.
When the velocity is zero, it signifies that there is no change in the position of the object, meaning it is stationary at that time.

Rate of Change

In calculus, the rate of change is a measure of how a quantity alters with another. For example, the rate of change of position with respect to time is described by velocity. It gives a dynamic view of how changes happen over time.
The importance of understanding the rate of change appears in various contexts:

• It's the basis for understanding motion and how different physical quantities interact.
• In practical terms, knowing the rate helps predict future behavior, such as predicting how long it will take for a car to reach a certain speed.

In the context of a particle's movement, if the rate of change, or derivative of the position function, is zero, it indicates that there is no change in position over that period. This is a critical observation when analyzing motion, as it confirms that the object is indeed at rest.

Position Function

The position function \( x(t) \) provides critical information about the location of an object at any given time \( t \). In essence, this function maps the time to the position, allowing us to visualize or calculate where an object is at specific instances.
Some key insights about the position function include:

• It's used to determine where an object is along its path of motion.
• The derivative of this function gives us vital motion characteristics like velocity.

When analyzing problems involving motion, like a particle's movement on a line, the position function is our starting point. An important aspect of this function occurs when its derivative (the velocity) is zero. At that point, the function remains unchanged, suggesting that the object hasn't moved—it's stationary during that interval. Therefore, understanding and interpreting the position function is crucial for any study of motion.