Two balls collide in a totally inelastic collision

Question

Two balls collide in a totally inelastic collision. The first ball has twice the mass of the second. The first ball has velocity v (to the right), and the second has velocity −v (to the left). What is the final velocity of the two balls after they collide? What was the % change in kinetic energy?

 

It's a homework problem worth 20 points and I'm not sure if my answer is correct. I said that both would have the same velocity because they are inelastic so i put m(v)+2m(-v)/2m*m= v3 and i have no idea how to find the % change in kinetic energy. 

 

Please help!! Its urgent!!

Answer 1

What you need to do is use KE = mv²/2 formula. In the beginning, both balls are moving with a speed of v, so total KE for the system is 2mv²/2 for the heavy one and mv²/2 for the light one, which gives you 3mv²/2 for the system in total.

After the balls collide, you'll have their common speed as ... I am not sure you found it correctly, by the way. With conservation of momentum, 2mv_i - mv_i = (2m+m)v_f, which means mv_i = 3mv_f, making v_f = v_i/3, only 1/3 of the original speed. 

So, the KE of the system after the collision is 3m*(v/3)²/2.

Once you square, you get mv²/6

 

So, you went from KE_i = 3mv²/2 to KE_f = mv²/6

Common denominator will give you 9mv²/6 for initial energy, then your change in KE is 8mv²/6.

If you are looking for the % of the change, you divide the change by the original

 

(4/3)/(3/2)=8/9 = 89% of energy was lost.

 

You can check with numbers if it helps:

let's say you have two balls: 1kg and 2 kg. The heavier is moving with a +5 m/s, the light - with - 5m/s

After they collide, their speed is 5/3 m/s.

You can check: 2*5 - 1*5 = (2+1) v, which gives you v = 5/3

 

Then let's find KE for each ball: 2 kg KE = 2*25/2 = 25 J, 1 kg KE = 1*25/2 = 12.5 J. Total energy is 37.5 J

 

After the collision, KE = (2+1)*(5/3)²/2 = 3*25/18=25/6 .

 

How much energy was lost? 37.5 - 25/6 = (225-25)/6 = 200/6 J.

 

What is that in relation to the original energy? (200/6)/37.5 = 0.89 = 89%