Physics question-confused

Question

A 120 kg mass is blown apart into a 80 kg piece and a 40 kg piece. After the blast, the two masses are moving apart with a relative velocity of 60 m/s. The velocity of the 80 kg mass is?

Hint: 60m/s=V1-V2

Answer 1

This problem is meant to illustrate the concept of conservation of linear momentum. Also, though this is by implication only, it spurs a discussion of frames of reference (hopefully inertial ones). We have the original 120 kg object and then it "explodes" into two pieces: one with an 80 kg mass and another with a 40 kg. mass which recede from each other with a relative velocity of 60 m/s.

First we will select as a frame of reference (again, this is implied by the wording of the problem) where the 120 kg. object is stationary. In addition, we will select a coordinate system, where the its remnants will recede along one of those axes, say the x-axis.

Now in this frame of reference, the total initial momentum, will be, because the object is initially stationary, zero. Now the x - component (as well as the y and z-components) of the initial momentum will be 0.

p_ix = 0

Now, the formula for total momentum after the explosion, with the 80 kg mass having a mass of m₈₀ = 80 kg with a velocity, v₁ and the 40 kg mass having a mass of m₄₀ = 40 kg and a velocity of v₂ is

p_fx = m₈₀v₁ + m₄₀v₂

(By the way, one question; if you were to divide the expression on the right, by the total mass of the objects, what would the expression indicate?)

Now using conservation of linear momentum along the x axis:

p_fx = p_ix

m₈₀v₁ + m₄₀v₂ = 0

Now solving for v₂ in terms of v₁

m₄₀v₂ = -m₈₀v₁

v₂ = -(m₈₀/m₄₀)v₁

v₂ = -2v₁

Now taking the expression of the velocity of the 80 kg object with respect to the 40 kg object,

v_relative = v₁ - v₂

v_{relative = 60 m/s}

and substituting our expression for v₂ with respect to v₁,

60 = v₁ - (-2v₁)=3v₁

v₁ = (60 m/s)/3 = 20 m/s and this would be the velocity of the 80 kg portion, if we take the relative velocity to be in the positive x direction.

the velocity of the 40 kg piece will be

60 = v₁ - v₂

v₂ = v₁ - 60 = (20 - 60) = -40 m/s in the opposite direction.

Answer 2

Use conservation of momentum, in addition to the given relative speed. The initial and final momenta are both zero. Note that the smaller mass must also be moving faster but in the opposite direction, in order for momentum to be conserved.

m₁ = 40 kg

m₂ = 80 kg

Be careful with signs when you set this up:

m₁v₁ + m₂v₂ = 0

v₁ - v₂ = 60

This gives us a system of 2 linear equations in 2 unknowns:

40v₁ + 80v₂ = 0

v₁ - v₂ = 60

Its solution is

v₁ = 40 m/s

v₂ = -20 m/s

The negative sign in v₂ means it is moving in the opposite direction to v₁.