Set 4 Problem number 11

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

Two objects collide and become embedded in one another.

What is their total momentum, and what will be the velocity of this system immeditely after collision, if one object is moving in the positive direction at 24 m/s and the other at 19 m/s in the negative direction, and if the mass of the first object is 6 kg and that of the second is 27 kg.

Solution

The momentum of the first is ( 24 m/s)( 6 kg)= 144 kg m/s, and that of the second is ( - 19 m/s)( 27 kg) = -513 kg m/s.

• The total momentum is therefore 144 kg m/s + -513 kg m/s =  -369 kg m/s.
• The total mass is easily found to be 33 kg. Since the total momentum after collision is the same as that before collision, we see that after collision we have a mass of 33 kg with momentum -369 kg m/s.
• Dividing we obtain velocity ( -369 kg m/s)/( 33 kg) = -11.18182 m/s.

Generalized Solution

The total momentum of a mass m1 moving at velocity v1 and a mass m2 moving at velocity v2 is

• pTot = m1 v1 + m2 v2.

By Newton's Third Law and the Impulse-Momentum Theorem this momentum will remain unchanged during collision.

After collision we wil have one object of mass m1 + m2 and momentum m1 v1 + m2 v2. The object will therefore have velocity

velocity = momentum / mass = (m1 v1 + m2 v2) / (m1 + m2).