Question

A car is moving at constant speed v on a parabolic path given by $y=\frac{x^2}{4}$ (as shown in figure). Coefficient of friction between road and car is 0.8
[take $g=10m/s^2$]

• A. As car moves from A to C friction on it decreases.
• B. As car moves from A to C friction on it first increases and then decreases
• C. Maximum speed at which car can move without skidding is 4 m/s
  
• D. If car is moving with 8 m/s at A, it will skid before reaching B.

Answer 1

Answer: (B), (C), (D)

The correct options are
B As car moves from A to C friction on it first increases and then decreases
C Maximum speed at which car can move without skidding is 4 m/s

D If car is moving with 8 m/s at A, it will skid before reaching B.

Rodius of curvature of parabola is

$R=\frac{{[1+{\left(\frac{dy}{dx}\right)}^2]}^{\frac{3}{2}}}{\frac{d^{2}y}{d{x}^2}},wherey=\frac{x^2}{4}$

So, $R=\frac{{(1+\frac{x^2}{4})}^{\frac{3}{2}}}{\frac{1}{2}}$

friction provides necessary centripetal force

So, $f=\frac{m{v}^2}{R},wheref=\mu mg$

as $v=\sqrt{\mu Rg}=\sqrt{\frac{\mu g{(1+\frac{x^2}{4})}^{\frac{3}{2}}}{\frac{1}{2}}}$

For v to be minimum x = 0

$v=\sqrt{0.8\times 10\times 2}=4m/s$