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Chapter 7: Problem 1

For each relationship below, indicate whether the two variables (indicated in bold) are proportional or inversely proportional: a) \(\quad \mathbf{y}=5 \mathbf{x}\) b) \(\quad \mathbf{p}=325 / \mathbf{v}\) c) For a given amount of force, if the mass of a car is doubled, the acceleration of the car is cut in half.

Short Answer

Expert verified
a) The variables 'y' and 'x' in the equation are proportional. b) The variables 'p' and 'v' in the equation are inversely proportional. c) The acceleration of the car is inversely proportional to the mass of the car.

Step by step solution

01

Analyze equation for proportionality

The equation \(\mathbf{y}=5 \mathbf{x}\) is in the form of a proportional relationship because 'y' and 'x' are multiplied together. A proportional relationship between two variables means that as one variable increases, the other variable increases at a constant rate.
02

Analyze equation for inverse proportionality

The equation \(\mathbf{p}=325 / \mathbf{v}\) is in the form of an inverse proportional relationship because 'p' is equal to a constant divided by 'v'. An inverse proportional relationship between two variables means that as one variable increases, the other variable decreases at a constant rate.
03

Analyze Acceleration-Force-Mass relationship

From Newton's second law of motion, acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. In this case, the statement 'For a given amount of force, if the mass of a car is doubled, the acceleration of the car is cut in half' implies that acceleration is inversely proportional to mass when a constant force is applied.

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Most popular questions from this chapter

a) Write the mathematical equation that relates the energy of a photon and its wavelength. b) Is the energy of a photon proportional or inversely proportional to \(\lambda\) ? $$ \begin{array}{ll} \hline \text { Region } & \text { Wavelength Range } \\ \hline \text { radiowave } & 3 \mathrm{~km}-30 \mathrm{~cm} \\ \text { microwave } & 30 \mathrm{~cm}-1 \mathrm{~mm} \\ \text { infrared (IR) } & 1 \mathrm{~mm}-800 \mathrm{~nm} \\ \text { visible (VIS) } & 800 \mathrm{~nm}-400 \mathrm{~nm} \\ \text { ultraviolet (UV) } & 400 \mathrm{~nm}-10 \mathrm{~nm} \\ \text { X-ray } & 10 \mathrm{~nm}-0.1 \mathrm{~nm} \\ \text { gamma ray } & <0.1 \mathrm{~nm} \\ \hline \end{array} $$ $$ \begin{array}{ccc} \hline \hline \begin{array}{c} \text { Wavelength } \\ (\mathrm{nm}) \end{array} & \begin{array}{c} \text { Frequency } \\ \left(10^{14} \mathrm{~s}^{-1}\right) \end{array} & \begin{array}{c} \text { Energy } \\ \left(10^{-19} \mathrm{~J}\right) \end{array} \\ \hline 333.1 & 9.000 & 5.963 \\ 499.7 & 6.000 & 3.976 \\ 999.3 & 3.000 & 1.988 \\ \hline \end{array} $$

The frequency \((f)\) of a wave is defined as the number of wavelengths per second which travel past a given point. a) For a wave traveling at a given speed, \(c\), how does the frequency depend on the wavelength, if at all? b) Provide a mathematical expression showing the relationship between \(f\), \(\lambda\), and \(c\) for a wave. (Hint: consider how you determined answers to CTQs 4 and \(4 b\) ).
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