Live practice

Principle. Combine the sound-speed formula with the ideal-gas relation P/ρ=RT/M.

At fixed T and composition, P/ρ is constant, so changing pressure alone does not change v. The formula gives v∝√T. Humid air has a lower mean molar mass than dry air because water vapour replaces heavier nitrogen and oxygen; lower M makes v larger.

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

The triple point is a unique, reproducible equilibrium of ice, water, and vapour at a specified pressure. Ordinary melting and boiling temperatures depend on pressure and impurities, so they are less exact standards.

The other physically distinguished fixed point is absolute zero, 0 K. The Celsius zero is the ice point, 0.01 K below the 273.16 K triple point; therefore T(K)=t(°C)+273.15.

A Fahrenheit degree is \(\frac{5}{9}\) of a kelvin. The triple-point reading is therefore 273.16×\(\frac{9}{5}\)=491.688 on an absolute Fahrenheit-sized scale (the Rankine scale).

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

Blowing above the paper raises air speed there, lowering static pressure; the larger pressure below lifts it. Narrow gaps at a tap require a larger speed for the same flow rate by continuity.

For viscous flow through a needle, Poiseuille's law gives Q=πr⁴ΔP/(8ηl), so radius has a fourth-power effect while thumb force changes pressure only linearly.

An outgoing jet carries forward momentum, so the vessel receives an equal backward reaction. Spin makes air speeds and pressures unequal on opposite sides of a ball; the resulting Magnus force bends the trajectory.

Principle. For a nearly circular satellite, gravity provides centripetal acceleration.

Principle. Addition requires like mathematical objects and compatible dimensions. Scalar multiplication of a vector is defined; ordinary multiplication of scalars is defined.

Each complete \(8\,\mathrm{s}\) cycle gives a net advance of \(2\,\mathrm{m}\). Four full cycles place the walker at \(8\,\mathrm{m}\) after \(32\,\mathrm{s}\); the next five forward steps then reach the pit.

Principle. Words such as large and small describe a ratio. A dimensional number alone changes when its unit changes.

Form the required totals explicitly and divide by the full number of observations.

The variance is nonnegative and the reported standard deviation is its positive square root.

Simplify the expression before substitution and preserve one-sided information.

Substitution into the original expression verifies the stated domain and final value.

Translate every condition into an equation and retain exact values throughout the conic calculation.

The last line satisfies the original conditions and supplies every requested part.