Try this problem useful for Physics Olympiad based on Relative motion.

**The Problem: Relative motion Problem**

A 40kg boy is standing on a plank of mass 160Kg. The plank originally at rest, is free to slide on a smooth frozen lake. The boy walks along the plank at a constant speed of 1.5m/s relative to the plank. What is the speed of the boy relative to the ice surface?

**Solution:**

Let the mass of the boy be (m_b) and that of the plank be (m_p). Let us consider the velocity of the boy with respect to the ice be (v_i), that of the plank with respect to ice be (v_i) and that of the boy with respect to the plank is (v_{bp}).

Now, using conservation of momentum,

$$m_bv_{bi}+(m_pv_{pi})=0$$

$$(40)v_{bi}+(160)v_{pi}=0$$

$$v_{bi}=-4v_{pi}….(i)$$

Also,

$$v_{bi}=v_{bp}+v_{pi}$$

From equation (i),

$$-4v_{pi}=v_{bp}+v_{pi}$$

$$-5v_{pi}=v_{bp}$$

$$v_{pi}=-\frac{1.5}{5}=0.3$$

Hence,

$$v_{bi}=1.2m/s$$

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