Tricky physics problem about impulses

Question

A projectile shot upwards, at the highest point explodes into three fragments. The first two fragments flied away with an angle of 90 degrees.

The mass of the first fragment is m1 = 9 kg, velocity v1 = 60 m/s, the mass of the second fragment m2 = 18 kg, velocity v2 = 40 m/s, the velocity of the third fragment is v3 = 300 m/s. Find the mass and the direction of the third fragment?

Answer 1

I do not think that this problem is tricky, or about impulses --

it is about conservation of momentum.

The momentum of the projectile at the time of explosion must equal the total sum

of the momenta of the three fragments.

Recall that momentum is a vector, and it is the product of mass and velocity.

Assuming that the projectile is shot directly vertically upward,

at the highest point it has velocity 0, hence momentum 0.

So the sum of the momenta of the three fragments must add up to 0.

So the momentum of the third fragment must be opposite the sum of the momenta of the first two.

The magnitude of the momentum of the first fragment is

m₁v₁ = 9×60 = 3×180 kg m/s

The magnitude of the momentum of the second fragment is

m₂v₂ = 18×40 = 4×180 kg m/s

As we are told that the two vectors form an angle of 90°,

their vector sum is the diagonal of the rectangle formed by the two momenta.

Consider either of the two right-angle triangles, whose hypotenuse is the vector sum.

Its sides are in relation 3:4.

By Pythagorean theorem the length of the hypotenuse is 5×180 kg m/s.

So the third fragment has momentum of magnitude 5×180 kg m/s.

Since its velocity is 300 m/s, its mass is

m3 = 5×180/300 = 3 kg.

Looking back at that right-angle triangle (using trigonometry),

its other two angles are approximately 53.1° and 36.9°.

As the third fragment flies in direction opposite the hypotenuse, it forms

angles 143.1° and 126.9° with the other two fragments.

So the final answer is that the third fragment has mass of 3 kg.

It flies away in the plane formed by the other two fragments.

It forms an angle of 126.9° with the first fragment, and 143.1° with the second segment.