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Asked by Prashant DIGHE | 17 Sep, 2019, 09:56: PM

Expert Answer

when the block reaches A from top of curved surface, all its potential energy is converted to kinetic energy

Hence its speed v_A at A :- mgh = (1/2) m v_A² or v_A = (2gh)^1/2 = (2×10×4)^1/2 =

begin mathsize 12px style square root of 80 end style

m/s

Since AB is smooth surface , velocity at B = v_B =

begin mathsize 12px style square root of 80 end style

m/s

In the path BC, due to friction, moving block is subjected to retardation μg = 0.1×10 = 1 m/s²

Hence when the block reahes C, its speed is calculated as,

v_C² = v_B² - (2 a S ) = 80 - (2×1×2 ) = 76 or v_C =

begin mathsize 12px style square root of 76 end style

m/s

Since collision with vertical wall at C is elastic, block returns back with same speed

begin mathsize 12px style square root of 76 end style

m/s

and again it is subjected to retardation 1 m/s².

Hence when the block reaches back at B, its speed v_B' is calculated as ( v_B')² = 76 - (2×1×2) , or v_B' =

begin mathsize 12px style square root of 72 end style

m/s

Then the returning block travels the smooth surface in backward direction, falls back

and reaches the point A with same speed

begin mathsize 12px style square root of 72 end style

m/s

Let us call the movement A to B, B to C, then backward direction C to B, B to A as one cycle.

At end of every cycle, the difference between square of initial speed and square of final speed is 8 m²/s² .

Speed and horizontal distance covered at end of every cycle is listed in the following table

Cycle No.	Initial speed (m/s)	Final speed (m/s	Cumulative Distance Covered ( m )
1	$begin mathsize 12px style square root of 80 end style$	$begin mathsize 12px style square root of 72 end style$	6
2	$begin mathsize 12px style square root of 72 end style$	$begin mathsize 12px style square root of 64 end style$	12
3	$begin mathsize 12px style square root of 64 end style$	$begin mathsize 12px style square root of 56 end style$	18
4	$begin mathsize 12px style square root of 56 end style$	$begin mathsize 12px style square root of 48 end style$	24
5	$begin mathsize 12px style square root of 48 end style$	$begin mathsize 12px style square root of 40 end style$	30
6	$begin mathsize 12px style square root of 40 end style$	$begin mathsize 12px style square root of 32 end style$	36
7	$begin mathsize 12px style square root of 32 end style$	$begin mathsize 12px style square root of 24 end style$	42
8	$begin mathsize 12px style square root of 24 end style$	$begin mathsize 12px style square root of 16 end style$	48
9	$begin mathsize 12px style square root of 16 end style$	$begin mathsize 12px style square root of 8 end style$	54