Final answer:
The distance formula d=1/2gt^2 shows joint variation involving the acceleration due to gravity and the square of time, while the other formulas show direct variation. Distance covered and magnitude of displacement will be the same for an athlete completing one lap in a race.
Step-by-step explanation:
The distance formulas d=50t, d=1/2gt^2, and d=r*t are all equations used in physics to describe motion under different conditions. The formula d=50t suggests a direct variation between distance and time, with a constant speed of 50 m/s. The second formula, d=1/2gt^2, represents the distance fallen under gravity over time t, showing joint variation because it involves both the acceleration due to gravity (g) and the square of the time (t^2). Finally, the formula d=r*t indicates a direct variation, where d is distance, r is the rate or speed, and t is time.
Among the options provided, only the formula d=1/2gt^2 illustrates a joint variation as it involves two variables (g and t) changing at the same time. In mechanics, the concept of joint variation is crucial when examining how one physical quantity changes in comparison to two or more other quantities.
Regarding the example cases provided, the distance covered and the magnitude of the displacement will be the same in case (b) An athlete completes one lap in a race, assuming the lap is completed along a circular track returning to the starting point without any deviation.