Answer to Question #135418 in Mechanics | Relativity for Nomo

Question

A 4 kg ball moving with a velocity of 100m/s collides with a 16 kg ball moving with a velocity of 40m/s. (i) in the same direction, (ii) in the opposite direction. Calculate the velocity of the balls in each case if they coalesce on impact, and the loss of energy resulting from the impact. State the principle used to calculate the velocity.

Answer 1

I) The same direction:

Relative velocity:

$v=100-40=60\frac{m}{s}$

And $v_1'-v_2'=60$

Applying conservation of momentum:

$m_1v_1+m_2v_2=m_1v_1'+m_2v_2'$

We have the system of equations:

$v_1'-v_2'=60$

$4v_1'+16v_2'=1040$

Substract from the second equation the first one. We get:

$20v_2'=800->v_2'=40\frac{m}{s}$

Then $v_1'=100\frac{m}{s}$

II) The opposite direction:

$v=100+40=140\frac{m}{s}$

And $v_1'+v_2'=140$

From the conservation of momentum"

$m_1v_1+m_2v_2=m_1v_1'=m_1v_1'+m_2v_2'$

The system of equations:

$v_1'+v_2'=140$

$4v_1'+16v_2'=1040$

Multiplying the first equation by 4 and substract it from the second one, we get

$12v_2'=480->v_2'=40\frac{m}{s}$

Then $v_1=100\frac{m}{s}$