---
title: "A group of students is investigating a non-linear elastic band. The magnitude of the restoring force \\( F \\) exerted by the band is modeled by the equation \\( F = kx + bx^2 \\), where \\( x \\) is the stretch distance, and \\( k \\) and \\( b \\) are positive constants. The students wish to experimentally determine \\( k \\) and \\( b \\), and then investigate whether mechanical energy is conserved when the band launches a cart of mass \\( M \\) along a horizontal track.  The students use a force sensor and a meterstick to measure the restoring force \\( F \\) at various stretch distances \\( x \\). Their data are shown in the table below. You may use the blank columns in the table to record any calculated values.  | Stretch Distance \\( x \\) (\\( \\text{m} \\)) | Force \\( F \\) (\\( \\text{N} \\)) |   |   | | :—: | :—: | :—: | :—: | | \\( 0.10 \\) | \\( 1.3 \\) |   |   | | \\( 0.20 \\) | \\( 3.2 \\) |   |   | | \\( 0.30 \\) | \\( 5.7 \\) |   |   | | \\( 0.40 \\) | \\( 8.8 \\) |   |   | | \\( 0.50 \\) | \\( 12.5 \\) |   |   |"
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url: "https://nerd-notes.com/ubq/117795/"
date_modified: "2026-08-04T07:57:03+00:00"
---

# A group of students is investigating a non-linear elastic band. The magnitude of the restoring force \( F \) exerted by the band is modeled by the equation \( F = kx + bx^2 \), where \( x \) is the stretch distance, and \( k \) and \( b \) are positive constants. The students wish to experimentally determine \( k \) and \( b \), and then investigate whether mechanical energy is conserved when the band launches a cart of mass \( M \) along a horizontal track.

The students use a force sensor and a meterstick to measure the restoring force \( F \) at various stretch distances \( x \). Their data are shown in the table below. You may use the blank columns in the table to record any calculated values.

| Stretch Distance \( x \) (\( \text{m} \)) | Force \( F \) (\( \text{N} \)) |   |   |
| :—: | :—: | :—: | :—: |
| \( 0.10 \) | \( 1.3 \) |   |   |
| \( 0.20 \) | \( 3.2 \) |   |   |
| \( 0.30 \) | \( 5.7 \) |   |   |
| \( 0.40 \) | \( 8.8 \) |   |   |
| \( 0.50 \) | \( 12.5 \) |   |   |

A group of students is investigating a non-linear elastic band. The magnitude of the restoring force \( F \) exerted by the band is modeled by the equation \( F = kx + bx^2 \), where \( x \) is the stretch distance, and \( k \) and \( b \) are positive constants. The students wish to experimentally determine \( k \) and \( b \), and then investigate whether mechanical energy is conserved when the band launches a cart of mass \( M \) along a horizontal track.

The students use a force sensor and a meterstick to measure the restoring force \( F \) at various stretch distances \( x \). Their data are shown in the table below. You may use the blank columns in the table to record any calculated values.

| Stretch Distance \( x \) (\( \text{m} \)) | Force \( F \) (\( \text{N} \)) |   |   |
| :---: | :---: | :---: | :---: |
| \( 0.10 \) | \( 1.3 \) |   |   |
| \( 0.20 \) | \( 3.2 \) |   |   |
| \( 0.30 \) | \( 5.7 \) |   |   |
| \( 0.40 \) | \( 8.8 \) |   |   |
| \( 0.50 \) | \( 12.5 \) |   |   |

![A side-view physical setup showing a horizontal track. At the left end of the track is a vertical post. A cart labeled M is on the track to the right of the post. The cart is attached to the post by a stretched elastic band. A horizontal arrow labeled x extends from the relaxed position of the band to the current position of the cart, indicating the stretch distance. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830223-bzA8md.jpg)

**Part a)** The students want to create a linear graph to determine the experimental values of \( k \) and \( b \). *(5 points)*

**Part b)** **Derive** an expression for the work \( W \) done by the elastic band on a cart as the band contracts from a maximum stretch distance \( D \) to its relaxed length (\( x = 0 \)). Express your answer in terms of \( k \), \( b \), \( D \), and fundamental constants. *(2 points)*

**Part c)** The students want to determine if mechanical energy is conserved when the band launches a cart of known mass \( M \) on a horizontal track. They attach the elastic band to a fixed post at one end of the track. **Describe** an experimental procedure the students could use to collect the necessary data to determine if mechanical energy is conserved. Assume the students have access to equipment commonly found in a high school physics laboratory. - List the additional equipment needed. - Describe the measurements to be made, including how the equipment would be used. - Include steps to reduce experimental error. *(3 points)*

**Part d)** **Explain** how the students should analyze the data collected in part (c) to determine whether mechanical energy is conserved during the launch. *(2 points)*

**Part e)** The students perform the experiment and calculate the initial theoretical elastic potential energy \( U \) stored in the band and the final kinetic energy \( K \) of the cart for several trials using different values of \( D \). They plot \( K \) as a function of \( U \) and draw a best-fit line through the data. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117795/*