Caroline calculated the wave speed by measuring the time t for the wave to travel 150 cm. The uncertainty in this distance is 2 cm. For the reading at a water depth of 15 cm, the time t is 8.3 s with an uncertainty 0.5 s.

(i) Show that the absolute uncertainty in the wave speed at this time is 1.3 cm s^–1.

(ii) On the graph opposite, draw the error bar for the data point corresponding to d=15 cm.

Caroline hypothesized that the wave speed c is directly proportional to the water depth d.

(i) On the graph opposite, draw a line of best-fit for the data.
(ii) Suggest if the data support this hypothesis.

Another student proposes that c is proportional to d^0.5.

State a suitable graph that can be plotted to test this proposal.

There is a systematic error in Caroline’s determination of the depth.

(i) State what is meant by a systematic error.

(ii) State how the graph in (c) would indicate that there is a systematic error.

(i)
fractional uncertainty in distance = \(\frac{2}{{150}}\) and
fractional uncertainty in time = \(\frac{0.5}{{8.3}}\); { (allow use of percentage uncertainty)
fractional uncertainty in speed \( = \frac{2}{{150}} + \frac{{0.5}}{{8.3}}( = 0.074\) or \(7.4\% )\);
absolute uncertainty =18×0.074;
=1.3 (cms^–1)

or

maximum = \(\frac{152}{{7.8}}\);
minimum = \(\frac{148}{{8.8}}\);

shows subtraction of maximum and minimum and division by 2;

(ii) error bars drawn as ±1.3;

(i) smooth curve within limits of all error bars;

(ii) a straight line cannot be drawn;
that goes through all the error bars / that goes through the origin;

c versus  \(\sqrt d \) / d^0.5 or c² versus d or lg c versus lg d or ln c versus ln d;
Allow as symbols or written in words.

(i) error that is identical for each reading / error caused by zero error in instrument / OWTTE;

(ii) graph will not go through origin / intercept non-zero;
graph will not be straight line/linear;