The potential energy of a particle of mass 1kg moving along x-axis is given by U(x)=[x^2/2-x]J. If total mechanical energy of the particle is 2J find its maximum speed.

							Ans- 2Speed will be max when kinetic energy is maximum so potential energy will be minimum so differentiating the function put it equal to 0  so we obtain minimum for x=0 now kin potential energy is 0 and kinetic energy will be 2 .mv^2/2 is equal to 2 m  is 1 kg  therefore v=2 .
						

							total mechanical energy = U+ K.E  to attain maximum speed the obect must have maximum K.E as K.E will be maximum , U has to be minimum (conservation of energy )given - U(x) = x²/2-xfor this polynomial to be minimum x has to be equal to zero [ U`(x) = (2-x)2x +x²/(2-x) =0 then x =0,4 there is minima at x=0]so putting x =0 iñ U(x) we get U=0 therfore total mechanical energy =2 = 0+K.E(max)then K.Emax =2 1/2 mv² =2 then v²=4therefore v=2m/s
						

							What is the denominator of function . I had taken the denominator as x-2 is this correct or not According to this answer coming out is x =0 minimum and x= 4 max for p.e Is this correct or not
						

							We know that the object accelerate till force is applied on it and it attains maximum velocity just after the force becomes zero...  SO we know the negative of  potential energy gradient is force..  So differentiate it and put energy gradient equal to 0..  So at X =1 the force Is 0, get the potential energy, subtract it from total energy to get kinetic energy and find its velocity it will be 5^1/2
						

							
U={square of the x divided by the 2 and substracted by x} so differentiate this we get x=1 substituting x=1 in the functio we get U=-1 by 2.
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							Potential energy needed to be zero for getting maximum speed..U(x)=[x^2/2 - x] i.e, force will remain constant for it therefore differentiating U(x) wrt d(x) we get x-1for maximum speed d(U(x))/d(x) =0 therefore x=1 T.E = K.E+ P.E2 = 1/2mv^2 + x^2/2 - x=>2 =1/2mv^2 +1/2-1=>2=1/2mv^2 -1/2=>5/2=1/2 v^2.                    (Since m=1)=>v=√5 m/s
						

							For speed to be maximum potential energy should be minimum. So, differentiate the x^2/2 - x and u will get x-1 on differentiating. Equate it to zero and solve for x. U will get x=1. Now, put x= 1 in the equation U(x) = x^2/2 - 2 to get the value of U which is 1/2. T.E= P.E + K.E. putting P.E = 1/2, K.E= 1/2*m*v^2 where m is given 1kg and finally T.E= 2J in the equation given above we will get v= √5m/s