Question

A bid of mud stuck to a bicycle's front wheel $(radius=30cm)$ detaches and is flung horizontally forward when it is at the top of the wheel's rotation $($directly above the wheel's hub$)$. The bicycle is moving forward at a speed $v=5\sqrt{3}m/s$ and the bicycle is rolling along without slipping. At the instant when the bit of mud hits the ground, how far $($in $m)$ is it from the lowermost point of the bicycle's front wheel? $($ignore air resistance$)$

• A. $3$
• B. $6$
• C. $4.5$
• D. $9$

Answer 1

The correct option is A $3$

As the bicycle is rolling without slipping, speed of mud when it detaches from it at topmost point $v=5\sqrt{3}+5\sqrt{3}=10\sqrt{3}m/s$
Mud will fall a vertical distance $=$ diameter of wheel $=h=2\times 30=60cm=0.6m$
Time taken to reach ground by mud $=t=\sqrt{\frac{2h}{g}}=\sqrt{\frac{2\times 0.6}{10}}=\frac{\sqrt{3}}{5}s$
Horizontal distance covered by mud $=x_1=vt=10\sqrt{3}\times \frac{\sqrt{3}}{5}=6m$
In the same time bicycle moves forward by distance $=x_2=vt=5\sqrt{3}\times \frac{\sqrt{3}}{5}=3m$
Distance between them $=x_1-x_2=6-3=3m$