Question

In outer space a rock with mass 7 kg, and velocity <3400, -2700, 2000> m/s, struck a rock with mass 17 kg and velocity <200, -290, 340> m/s. After the collision, the 7 kg rock's velocity is <3000, -2100, 2300> m/s.
What is the final velocity of the 17 kg rock?
What is the change in the internal energy of the rocks?

Answer 1

Part A -

Solve this problem using unit vector notations. (i, j, k)

Suppose, mass, m1 = 7 kg

its initial velocity, u1 = (3400i - 2700j + 2000k) m/s

its final velocity after collision, v1 = (3000i -2100j + 2300k) m/s

also,

m2 = 17 kg

its initial velocity, u2 = (200i -290j + 340k) m/s

Suppose, its final velocity after collision, v2 = (ai + bj + ck) m/s

Total initial momentum -

Pi = m1*u1 + m2*u2

= 7*(3400i - 2700j + 2000k) + 17*(200i -290j + 340k)

= (27200i - 23830j + 19780k) kg*m/s

Total final momentum -

Pf = m1*v1 + m2*v2

= 7*(3000i - 2100j + 2300k) + 17*(ai + bj + ck)

= [ (21000+17a)i + (-14700 + 17b)j + (16100 + 17c)j ]

According to conservation of momentum -

Pi = Pf

=> (27200i - 23830j + 19780k) = (21000+17a)i + (-14700 + 17b)j + (16100 + 17c)j

Equalize i, j and k components -

27200 = 21000 + 17a

=> a = 365

-23830 = -14700 + 17b

=> b = -537

19780 = 16100 + 17c

=> c = 216

Therefore, final velocity of rock 17kg = v2 = (ai + bj + ck) m/s

= (365i -537j + 216k) m/s

Part B -

|u1| = sqrt[3400^2 + (-2700)^2 + 2000^2] = 4780 m/s

|u2| = sqrt[200^2 + (-290)^2 + 340^2] = 490 m/s

|v1| = sqrt[3000^2 + (-2100)^2 + 2300^2] = 4324 m/s

|v2| = sqrt[365^2 + (-537)^2 + 216^2] = 684 m/s

Total initial kinetic energy = (1/2)*m1*u1^2 + (1/2)*m2*u2^2

= 0.5*7*4780^2 + 0.5*17*490^2 = 82010250 J

Total final kinetic energy = (1/2)*m1*v1^2 + (1/2)*m2*v2^2

= 0.5*7*4324^2 + 0.5*17*684^2 = 69416192 J

So, change in internal energy of the rocks = Total initial kinetic energy - Total final kinetic energy

= 82010250 - 69416192 = 12594058 J (Answer)