A block of mass m and a block of mass 3m are used to compress a massless spring

Question

A block of mass m and a block of mass 3m are used to compress a massless spring with spring constant k. The blocks are not connected to the spring. The blocks start at rest on a frictionless surface, and the spring between them is initially compressed by an amount D from its equilibrium. The blocks are then released, and the spring launches the blocks in opposite. (a) Find the initial accelerations of the blocks. (b) Can we use the kinematic equation v^2 = 2a_x Delta x to find the final velocities of the blocks? Explain. (c) Is this process elastic? (d) Find the final velocities of the two blocks.

Explanation / Answer

a) let m1 = m

m2 = 3*m

acceleration of m1, a1 = F/m1

= k*D/m

acceleration of m2, a2 = F/m2

= k*D/(3*m)

b) No. because the acceleration of the object are not constant As the spring can not exert a constant force.

c) Yes. Initial elastic potential energy is equal to final kinetic energy.

d)

let v1 and v2 are the soeed of m1 and m2 when they leave the spring.

Apply conservation of momentum

m1*v1 = m2*v2

m*v1 = 3*m*v2

==> v2 = v1/3

Apply conservation of momentum

(1/2)*k*D^2 = (1/2)*m1*v1^2 + (1/2)*m2*v2^2

k*D^2 = m*v1^2 + 3*m*(v1/3)^2

k*D^2 = m*v1^2*(1 + 1/3)

k*D^2 = (4/3)*m*v1^2

==> v1^2 = 3*k*D^2/(4*m)

v1 = sqrt(3*k*D^2/(4*m))

= (D/2)*sqrt(3*k/m) <<<<<<-----------Answer

v2 = v1/3

= (D/6)*sqrt(3*k/m) <<<<<<-----------Answer

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