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 23 m/s and the other at 11 m/s in the negative direction, and if the mass of the first object is 28 kg and that of the second is 13 kg.

Solution

The momentum of the first is ( 23 m/s)( 28 kg)= 644 kg m/s, and that of the second is ( - 11 m/s)( 13 kg) = -143 kg m/s.

• The total momentum is therefore 644 kg m/s + -143 kg m/s =  501 kg m/s.
• The total mass is easily found to be 41 kg. Since the total momentum after collision is the same as that before collision, we see that after collision we have a mass of 41 kg with momentum 501 kg m/s.
• Dividing we obtain velocity ( 501 kg m/s)/( 41 kg) = 12.21951 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).