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A block of mass 10 kg moving with a velocity of 1 m/s collides with a spring of force constant 1000 N/m. Calculate the compression of the spring at the moment when kinetic energy of the block is equal to one-fourth of the elastic potential energy of the spring.

Attempt:-

Step 1: Determine initial Kinetic Energy of block

Step 2: Using above value of K.E., determine maximum compression of spring using Law of Conservation of Mechanical Energy

Step 3: Determine acceleration for the maximum compression using 3rd Kinematic equation

Step 4: Obtain two equations with two unknowns (v & x):

- First: v

^{2}= u

^{2}+ 2ax (3rd Kinematic equation)

- Second: 1/4*1/2*k*x

^{2}= 1/2*m*v

^{2}(from question statement)

Step 5: Substitute v

^{2}from First equation in the Second and solve 25x

^{2}+ 10x - 1 = 0 for x.

Problem:-

The above question is a 4 marks exam question which gives me 9 minutes to solve it. In addition to the 4 steps preceding it, Step 5's quadratic equation doesn't yield whole number roots. To be precise they are x=.09 & x=-.48.

There must be a shorter procedure.

Would appreciate advise.

Regards,

wirefree