The velocity-displacement graph describing the motion of bicycle is shown in the figure. The acceleration-displacement graph of the bicycle's motion is best described by :
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Use \(a = \frac{dv}{dt} = v\frac{dv}{dx}\); v is linear in x up to 200 m and constant afterwards.
Reason by shape. Chain rule: \(a = \frac{dv}{dt} = \frac{dv}{dx}\cdot\frac{dx}{dt} = v\frac{dv}{dx}\). In 0 to 200 m, \(\frac{dv}{dx}\) is a constant and v is a linear function of x, so a is a linear (straight-line) function of x with positive slope and a non-zero intercept at x = 0 (a = 2 m/s2 there, since \(v = 10\) and \(\frac{dv}{dx} = 0.2\)). This eliminates the curved graphs (options 2 and 3). In 200 to 400 m the speed is constant, so \(a = 0\); this eliminates option (4), which shows a constant non-zero acceleration there (and option 3). Only option (1) (straight line rising to x = 200 m, then zero acceleration) fits. Answer: (1)
