Momentum in physics on an explosion

Question

a plastic sphere containing an explosive charge, rolling at 4 m/s undergoes a two dimensional explosion into 3 fragments of equal mass. when viewed from above, one fragment moves at 6 m/s at an angle of 30 degrees to the right of the original line of motion, one fragment moves at 5 m/s at 45 degrees to the left of the original line of motion.

 

2)what is the velocity of the third fragment?

 

1)Show that the velocity of the center of mass of the system is the same before and after the explosion.

Answer 1

I will set this up for you and you can crunch out the numbers.

 

v = initial speed of sphere = 4 m/s, assumed to be in +x direction

m = initial mass of sphere, which is unknown, and it will drop out of the problem anyway, as you will see below

m/3 = mass of each of the three fragments after the explosion

u₁ = speed of fragment 1 = 6 m/s

θ₁ = angle of u₁ = -30º

u₂ = speed of fragment 2 = 5 m/s
θ₂ = angle of u₂ = 45º

u₃ = speed of fragment 3 = ?
θ₃ = angle of u₃ = ?

 

The explosion involves internal forces, so linear momentum is conserved.

 

Along x-axis:

 

mv = (m/3)u₁cosθ₁ + (m/3)u₂cosθ₂ + (m/3)u₃cosθ₃

 

Note m cancels out.  This simplifies to 

 

(i)   3v = u₁cosθ₁ + u₂cosθ₂ + u₃cosθ₃

 

Along y-axis:

 

0 = (m/3)u₁sinθ₁ + (m/3)u₂sinθ₂ + (m/3)u₃sinθ₃

 

This simplifies to

 

(ii)   0 = u₁sinθ₁ + u₂sinθ₂ + u₃sinθ₃

 

Solve for the two unknowns u₃ (speed) and θ₃ (direction) from (i) and (ii).  That will give you velocity of fragment 3.  Be careful that you place θ₃ in the correct quadrant.

 

Let v_c = velocity of center of mass

 

v_cxi = initial speed of center of mass along x-direction, v_cyi = initial speed of center of mass along y-direction

 

v_cxi = (mv)/m = v

 

v_cyi = 0

 

v_cxf = final speed of center of mass along x-direction, v_cyf = final speed of center of mass along y-direction

 

v_cxf = [(m/3)u₁cosθ₁ + (m/3)u₂cosθ₂ + (m/3)u₃cosθ₃] / m = (1/3) (u₁cosθ₁ + u₂cosθ₂ + u₃cosθ₃)

 

Similarly,

 

v_cyf = (1/3) (u₁sinθ₁ + u₂sinθ₂ + u₃sinθ₃)

 

Crunch out the numbers and compare v_cxi to v_cxf, and v_cyi to v_cyf.