---
title: "Students set up an experiment with a model rocket in a wind tunnel. The wind tunnel is programmed to produce a time-varying force that accelerates the rocket along a horizontal, low-friction guide wire. The acceleration \\(a\\) of the rocket model varies with time \\(t\\) according to the equation \\(a(t) = c\\sqrt{t}\\), where \\(c\\) is an unknown positive constant and \\(t \\ge 0\\). The rocket starts from rest at \\(x = 0\\) when \\(t = 0\\)."
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url: "https://nerd-notes.com/ubq/117804/"
date_modified: "2026-08-04T07:57:18+00:00"
---

# Students set up an experiment with a model rocket in a wind tunnel. The wind tunnel is programmed to produce a time-varying force that accelerates the rocket along a horizontal, low-friction guide wire. The acceleration \(a\) of the rocket model varies with time \(t\) according to the equation \(a(t) = c\sqrt{t}\), where \(c\) is an unknown positive constant and \(t \ge 0\). The rocket starts from rest at \(x = 0\) when \(t = 0\).

Students set up an experiment with a model rocket in a wind tunnel. The wind tunnel is programmed to produce a time-varying force that accelerates the rocket along a horizontal, low-friction guide wire. The acceleration \(a\) of the rocket model varies with time \(t\) according to the equation \(a(t) = c\sqrt{t}\), where \(c\) is an unknown positive constant and \(t \ge 0\). The rocket starts from rest at \(x = 0\) when \(t = 0\).

![A horizontal straight line representing a guide wire. On the guide wire is a simple model rocket pointing to the right. The rocket and wire are enclosed in a large rectangular box open at both the left and right ends, representing a wind tunnel. Wavy arrows pointing to the right enter the left side of the wind tunnel. A text label 'Wind' is placed near the wavy arrows. The rocket is labeled 'Model Rocket'. The horizontal line is labeled 'Guide Wire'. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830238-LFB5LO.jpg)

**Part a)** **Derive** an expression for the theoretical velocity \(v(t)\) of the rocket model as a function of time \(t\) and the constant \(c\).

**Part b)** The students are provided with the wind tunnel, the rocket model, the guide wire, and standard physics lab equipment. **Describe** an experimental procedure the students could use to collect the necessary data to determine the constant \(c\). In your procedure, include: - The equipment needed and how it will be used - The quantities to be measured - Steps to reduce experimental error

**Part c)** The students conduct the experiment and obtain the velocity of the rocket at one-second intervals, as shown in the table below. | \(t \text{ (s)}\) | \(v \text{ (m/s)}\) |           |           | |-------------------|---------------------|-----------|-----------| | 0.0               | 0.0                 |           |           | | 1.0               | 0.42                |           |           | | 2.0               | 1.10                |           |           | | 3.0               | 2.12                |           |           | | 4.0               | 3.15                |           |           | **Identify** which quantities should be graphed to yield a straight line whose slope could be used to calculate a value for \(c\). Vertical axis: Horizontal axis:

**Part d)** **Calculate** the values needed to graph the straight line and record them in the empty columns of the data table in part (c). Then, **plot** the data on the provided grid. Scale and label all axes, including units. **Draw** a straight line that best represents the data.

**Part e)** Using your straight line from part (d), **calculate** an experimental value for the constant \(c\).

**Part f)** Using a trapezoidal approximation and only the data points in the table, **calculate** an estimate for the total distance the rocket traveled from \(t = 0\) to \(t = 4.0 \text{ s}\). **Show** your calculations.

**Part g)** **Indicate** whether the trapezoidal approximation in part (f) overestimates or underestimates the actual distance traveled by the rocket model, assuming the theoretical model \(a(t) = c\sqrt{t}\) perfectly describes the motion. - [ ] Overestimates - [ ] Underestimates **Justify** your answer using physical or mathematical principles.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117804/*