You have been asked to design a \"ballistic spring system\" to measure the speed

Question

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass m is fired into a block of mass M. The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is k. The opposite end of the spring is anchored to a wall. The spring's maximum compression d is measured.

(a) Find an expression for the bullet's initial speed in terms of m, M, k, and d.

___________

(b) What was the speed of a 4.9 g bullet if the block's mass is 2.0 kg and if the spring, with k = 47 N/m, was compressed by 10 cm?
________ m/s

(c) What fraction of the bullet's energy is "lost"?

__________

___________

Explanation / Answer

let the initial speed of bullet v1 and speed of block after strike be v2

==> mv1 = (M+m)v2

==> v1= (M+m)v2/m

now conserving enrgy

0.5*(M+m)v2^2= 0.5*Kd^2

==> v2= sqrt(Kd^2/(M+m))

==> v1= (M+m)v2/m=(M+m)* sqrt(Kd^2/(M+m))/m= sqrt(Kd^2*(M+m)/m^2)

2) putting the values

v1= sqrt(47*0.1^2*2/(0.0049)^2)=197.86 m/s

3) final energy = 0.5*K*d^2=0.5*47*0.1^2= 0.235 J

initial energy= 0.5*mv1^2=0.5*(0.0049)*(197.86)^2= 95.91 J

so most of the energy is lost

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