MODEL #4 CUT-A4

Square Relationship

\( y = kx^2 \)

Independent Variable (x) Dep. (y)

SCENARIO CUT-B4

Lab: Kinetic Energy

Calculating kinetic energy \( (K) \) of a sliding hockey puck with a mass of \( 2.0\text{ kg} \) as it flies at different velocities \( (v) \).

Model: \( K = v^2 \)

DATA TABLE CUT-C4

MODEL #5 CUT-A5

Square Root Relationship

\( y = k\sqrt{x} \)

Independent Variable (x) Dep. (y)

SCENARIO CUT-B5

Lab: Pendulum Period

Measuring oscillation period \( (T) \) of a heavy brass pendulum as its suspension wire length \( (L) \) is extended.

Model: \( T = 2.0\sqrt{L} \)

DATA TABLE CUT-C5

Interactive Card Activity • Cut-and-Align Guide • Physics Graphing Sequence

Page 1 • Types of Graph - Summary

Types of graph - Summary

PHYSICS CORE SYSTEM

Analyzing Curves in Action: This section transitions into nonlinear behaviors. Some physics variables drop rapidly as they scale, while others rise exponentially. Study the concise scientific readings, theoretical models, and pre-drawn plots.

03

Inverse Proportionality

An inverse relationship occurs when one variable increases while the other decreases proportionally. Newton's Second Law demonstrates this: if you push carts of different masses with a constant force, the acceleration decreases as mass increases (\( a = F/m \)). Tripling the mass reduces acceleration to one-third of its original value. When plotted, this data forms a curving hyperbola that approaches the axes but never touches them.

Model: \( y = \frac{k}{x} \)  |  Shape: ____________________________

If mass triples, acceleration drops to ___________ of its original value.

Partner Share: Summarize 2 key points to share with your partner:

a (m/s²)

m (kg)

04

Square Relationship (Quadratic)

A square relationship represents a power law where the dependent variable is proportional to the square of the independent variable. In kinetic energy experiments, a vehicle's energy scales with the square of its speed (\( KE = \frac{1}{2}mv^2 \)). Doubling the speed of the vehicle quadruples its kinetic energy, while tripling speed increases energy ninefold. This yields an upward-curving parabola starting at the origin.

Model: \( y = kx^2 \)  |  Shape: _______________________________

If velocity triples, kinetic energy will scale by a factor of __________ .

Partner Share: Summarize 2 key points to share with your partner:

K (J)

v (m/s)

Page 2 • Types of Graph - Summary

Types of graph - Summary

PHYSICS CORE SYSTEM

05

Square Root Relationship

A square root relationship is the inverse of a square relationship, where the dependent variable is proportional to the square root of the independent variable. For instance, the period of a simple pendulum is proportional to the square root of its suspension length (\( T \propto \sqrt{L} \)). To double the time it takes for a pendulum to swing, the string length must be quadruled. This curves upward from the origin and slowly flattens out as length stretches.

Model: \( y = k\sqrt{x} \)  |  Shape: __________________________

To double the swing period, the length must increase by a factor of ____________ .

Partner Share: Summarize 2 key points to share with your partner:

T (s)

L (m)

STUDENT REFERENCE MATRIX

| | Linear |

|

| | Inverse |

|

| | Square |

|

| | Square Root |

|

|

Quick Comprehension Check

A lab team plotting acceleration vs. mass got a curving hyperbola. They want to find the applied force, which is the constant \( F \) in \( a = F(1/m) \).

Q1: What should they plot on the X-axis to linearize? ______________________________________________

Q2: What will the slope of the linearized graph equal? ______________________________________________

Page 3 • Types of Graph - Summary

1. Direct proportionality means doubling one variable exactly doubles the other.

2. The slope represents a physical constant of the system, such as electrical resistance \( R \).

V (V)

I (A)

Teacher Key • Page 1 • Types of Graph Summary

Teacher Answer Key

Types of Graph - Summary

PHYSICS KEY PACKET

Analyzing Curves in Action: This section transitions into nonlinear behaviors. Some physics variables drop rapidly as they scale, while others rise exponentially. Study the concise scientific readings, theoretical models, and pre-drawn plots.

03

Inverse Proportionality

An inverse relationship occurs when one variable increases while the other decreases proportionally. Newton's Second Law demonstrates this: if you push carts of different masses using a constant force, the acceleration decreases as mass increases (\( a = F/m \)). Tripling the mass reduces acceleration to one-third of its original value. When plotted, this data forms a curving hyperbola that approaches the axes but never touches them.

Model: \( y = \frac{k}{x} \)

Shape: Hyperbola (Downward Curve)

If mass triples, acceleration drops to one-third (1/3) of its original value.

Partner Share: Summarize 2 key points to share with your partner:

1. As the independent variable increases, the dependent variable decreases at a scaling rate.

2. The curve is an asymptotic hyperbola which never crosses either axis.

a (m/s²)

m (kg)

04

Square Relationship (Quadratic)

A square relationship represents a power law where the dependent variable is proportional to the square of the independent variable. In kinetic energy experiments, a vehicle's energy scales with the square of its speed (\( KE = \frac{1}{2}mv^2 \)). Doubling the speed of the vehicle quadruples its kinetic energy, while tripling speed increases energy ninefold. This yields an upward-curving parabola starting at the origin.

Model: \( y = kx^2 \)

Shape: Upward-Curving Parabola

If velocity triples, kinetic energy will scale by a factor of 9 (nine) .

Partner Share: Summarize 2 key points to share with your partner:

1. The dependent variable changes in proportion to the square of the independent variable.

2. Tripling the independent variable results in a 9x increase in the dependent variable.

K (J)

v (m/s)

Teacher Key • Page 2 • Types of Graph Summary

Teacher Answer Key

Types of Graph - Summary

PHYSICS KEY PACKET

05

Square Root Relationship

A square root relationship is the inverse of a square relationship, where the dependent variable is proportional to the square root of the independent variable. For instance, the period of a simple pendulum is proportional to the square root of its suspension length (\( T \propto \sqrt{L} \)). To double the time it takes for a pendulum to swing, the string length must be quadruled. This curves upward from the origin and slowly flattens out as length stretches.

Model: \( y = k\sqrt{x} \)

Shape: Upward Curve (Gradually Flattening)

To double the swing period, the length must increase by a factor of 4 (four) .

Partner Share: Summarize 2 key points to share with your partner:

1. The dependent variable increases rapidly initially, then slows down as X increases.

2. Doubling the output (period) requires quadrupling (4x) the input parameter (length).

T (s)

L (m)

COMPLETED REFERENCE MATRIX

Answers

Quick Comprehension Check (Solved)

A lab team plotting acceleration vs. mass got a curving hyperbola. They want to find the applied force, which is the constant \( F \) in \( a = F(1/m) \).

Q1: What should they plot on the X-axis to linearize? Plot \( 1/m \) (inverse mass) on the X-axis.

Q2: What will the slope of the linearized graph equal? The slope of the line will equal the applied force, \( F \).

Teacher Key • Page 3 • Types of Graph Summary