Problem 100 - Calculus Volume 1

Question

Use the following table, which shows the height \(h\) of the Saturn \(\mathrm{V}\) rocket for the Apollo 11 mission \(t\) seconds after launch. $$\begin{array}{|l|l|} \hline \text { Time (seconds) } & \text { Height (meters) } \\ \hline 0 & 0 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 13 \\ \hline 4 & 25 \\ \hline 5 & 32 \\ \hline \end{array}$$ What is the physical meaning of \(h^{\prime}(t) ?\) What are the units?

Step-by-Step Solution

Answer

The derivative \(h'(t)\) represents the rocket's velocity in m/s.

Step 1: Understanding the Notation

The notation \(h'(t)\) represents the derivative of the height function \(h(t)\) with respect to time \(t\). This derivative tells us how the height of the rocket is changing over time, which is basically its velocity.

Step 2: Physical Meaning

The physical meaning of \(h'(t)\) in the context of the rocket is that it represents the velocity of the rocket at a given time \(t\). Velocity is the rate of change of position, or in this case, the rate of change of height with respect to time.

Step 3: Determining Units

Since \(h(t)\) is measured in meters and \(t\) is measured in seconds, the derivative \(h'(t)\), which is the rate of change of height, has the units of meters per second (m/s). This is because it indicates how many meters the rocket ascends per second.

Key Concepts

Rate of Change

In the study of calculus, the rate of change is a fundamental concept. It describes how one quantity changes in relation to another. For our rocket example, the rate of change refers to how the height of the Saturn V changes as time progresses.

• In simple terms, it measures the steepness or slope of a curve, in this case, the height of the rocket over time.
• Mathematically, the rate of change is represented by the derivative. Here, it's noted as \(h'(t)\).
• This derivative tells us the velocity of the rocket, or how quickly its height is increasing at any point in time.

Understanding rate of change lets us predict how a system behaves over small intervals. By knowing \(h'(t)\), we can anticipate how the Saturn V's height adjusts from one second to the next. This insight is essential for tasks like anticipating when a rocket will reach a certain altitude.

Velocity

Velocity in physics is a vector quantity that refers to the speed of an object in a specified direction. For the Saturn V rocket, velocity indicates both how fast it is ascending and in which direction (upwards).

• It is derived from the rate of change of the rocket's height over time, using the derivative \(h'(t)\).
• Because velocity considers direction, it's important to note that if a rocket were descending, its velocity would be negative.

In our scenario, the velocity or \(h'(t)\) is expressed in meters per second. This reading is crucial for understanding the effectiveness and performance of the rocket during its ascent. By analyzing changes in \(h'(t)\), engineers can gauge how efficiently the rocket climbs and make necessary adjustments.

Units of Measurement

Units of measurement help us quantify and communicate observations consistently. For the Saturn V rocket, the height \(h(t)\) is measured in meters while time \(t\) is measured in seconds.

• The derivative \(h'(t)\), indicating velocity, has units of meters per second (m/s).
• This indicates the rocket's vertical speed; for example, at a given second, it's telling us how many meters the rocket travels upward per second.

Having a clear understanding of units is essential for precise scientific calculations and communications. They ensure that everyone interprets the measurements in the same way, avoiding confusion. Here, the meters per second (m/s) unit directly links to the physical concept of velocity, bridging theoretical math with tangible physical motion.