I posted two problems, if you could please help me and show me how they are done

Question

I posted two problems, if you could please help me and show me how they are done I would greatly appreciate it! Thanks! #1.) A 6-g bullet is fired into a 1.1-kg block that is resting on a frictionless surface and is attached to a spring with spring constant 371 N/m. After impact, the block with the embedded bullet slides 5 cm across the table, compressing the spring, before coming momentarily to rest. What was the initial speed of the bullet? (Practise how to derive an algebraic expression for the speed of the bullet before calculating the final result.) #2.)Calculate the ratio of earth's rotational kinetic energy due to its rotation around its axis and the translational kinetic energy due to its motion around the sun. Consider the earth to be a homogeneous solid sphere.

Explanation / Answer

bullet = 6.00g =0.006kg=1.106kg

now... conserving momentum let the initial velocity of bullet be v1 and final common velocity be v2... then

total mass of the bullet and cart=1.106*v2=0.006*v1 (i)

now ke=change in potential enrgy of spring

=0.5*1.106*(v2)^2=0.5*371*0.05^2

thusv2=0.91m/s using equation (i)

thus velocity of bullet=168.8 m/s

2.Ec=2/5m x r^2 x omega^2

both energies are kinetic rotational,`cos the earth turns around the sun and around itself
omega is the rotational speed of the earth,
=463.8km/s around itself
=29799.6km/s around the sun
earth radius=6000km
earth orbit radius=150,000,000km
the ratio is[2/5m x v(of roation)^2]/0.5mv^2=3.1*10^110/0.5*=4.845*10^-3

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