Live practice

A gas that extinguishes a flame and is released by an acid-carbonate reaction is carbon dioxide. The calcium compound is therefore calcium carbonate.

The carbon dioxide would also turn lime water milky due to formation of calcium carbonate.

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

A profile that depends on x−vt translates rigidly in +x; dependence on x+vt translates in −x. Therefore all three have the required travelling form.

Expression (a) moves in +x, though it is an unbounded profile rather than a localised pulse. Expressions (b) and (c) move in −x but exist only where their logarithm or denominator is defined. Mathematical singularities or nonlocalisation do not remove the translation property on a valid domain.

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

Take the midpoint as x=0, so A is x=−\(5\,\mathrm{cm}\) and B is x=+\(5\,\mathrm{cm}\). In SHM, a=−ω²x and F=ma, while velocity sign follows the stated direction.

(a) At A: v=0, a positive, F positive. (b) At B: v=0, a negative, F negative. (c) At the midpoint toward A: v negative, a=0, F=0.

(d) x=+\(3\,\mathrm{cm}\) toward A: v,a,F all negative. (e) x=−\(2\,\mathrm{cm}\) toward B: v,a,F all positive. (f) x=+\(1\,\mathrm{cm}\) toward A: v,a,F all negative.

Principle. At fixed amount and volume an ideal gas has T proportional to P, so calibrate each reading against 273.16 K.

Real gases depart slightly and differently from ideal behaviour. Repeat at successively lower gas pressures and extrapolate each result to zero pressure; both then approach the thermodynamic temperature.

Principle. For each wire use ΔL=FL/(AY). The upper wire supports both masses; the lower supports only the lower mass.

Principle. Use circular-orbit dynamics with total enclosed mass and convert light-years to metres.

Principle. The cross product magnitude gives base area and its direction is normal to that base; dotting with the third edge selects perpendicular height.

Principle. Track the mechanical energy of the stated system and use conservative-work and orbital-energy relations.

Convert speed to SI units and apply v²=u²+2as and v=u+at.

The new unit equals one light-second, so only the travel time in seconds is required.