Finding speed of objects after elastic collison

[joe426]

Homework Statement

Homework Equations

1/2m₁v₁² + 1/2m₂v₂² = 1/2m₁v_1f + 1/2m₂v_2f
m₁v₁ + m₂v₂ = m₁v_1f + m₂v_2f
v₁ - v₂ = -(v_1f - v_2f)

The Attempt at a Solution

So I solved the momentum of conservation for the final velocity of object 1. I then plug that equation into the third equation listed above, as v_1f. But the thing is the equation i get for v_1f has v_2f in it and that is what I am solving for and this is where I am stuck.

 

[joe426]

I don't think any of the velocity equals 0 at any point. Initially they are moving in the same direction. Then since it says its perfectly elastic, one object will bounce back and have a velocity in the opposite direction. I think the 60g tennis ball will bounce back in the negative direction but I still can't figure out how to find the velocities after the collision :(

 

[ehild]

Show your whole work please.

ehild

 

[joe426]

First use conservation of momentum
( m₁v₁² + m₂v₂² - m₂v_2f² ) / m₁ = v_1f²

Then plug this v_1f into this helper equation, the third equation listed in first post.
v_2f = v₁ - v₂ + [( m₁v₁² + m₂v₂² - m₂v_2f² ) / m₁ ]^1/2


And I'm stuck because I can't get both v_2f to one side so i can solve for it. then once i get v_2f i plug it back into the first equation, [( m₁v₁² + m₂v₂² - m₂v_2f² ) / m₁ ], and solve for v_1f

 

[ehild]

Use the last two equations. Plug in the numbers first. Express v_1f from the third equation.

ehild

 

[ehild]

1/2m₁v₁² + 1/2m₂v₂² = 1/2m₁v_1f + 1/2m₂v_2f
m₁v₁ + m₂v₂ = m₁v_1f + m₂v_2f
v₁ - v₂ = -(v_1f - v_2f)

Use the last two equations. Plug in the data first, express v_if form the last equation, and substitute into the second one.

ehild

 

[joe426]

v_1f = -v₁ + v₂ - v_2f

Plugged that into,
(m₁v₁ + m₂v₂ - m₁v_1f) / m₂ = v_2f

v_2f = -.368m/s

 

[ehild]

v_1f = -v₁ + v₂ - v_2f

Check it.

 

[joe426]

Check it.

v1f = -v1 + v2 - v2f is really v1f = -v1 + v2 + v2f. ..

(m1v1 + m2v2 - m1v1f) / m2 = v2f turns into (m1v1 + m2v2 + m1v1f) / m2 = v2f

so v2f = .307m/s

 

[ehild]

v1f = -v1 + v2 - v2f is really v1f = -v1 + v2 + v2f. ..

(m1v1 + m2v2 - m1v1f) / m2 = v2f turns into (m1v1 + m2v2 + m1v1f) / m2 = v2f

Check.

ehild

 

[ehild]

Plug in the numbers for v₁ and v₂: v_1f = v_2f - 1.35.
Plugging in the data into the equation m₁v₁+m₂v₂=m₁v_1f+m₂v_2f and also substituting v_2f - 1.35 for v_1f: 0.2535 = 0.06(v_2f - 1.35)+0.09v_2f.
Expand and collect the terms with v_2f, solve.

ehild

 

[Munzi5]

Its 3.75 or 1.35?

 

[ehild]

Its 3.75 or 1.35?

What is 3.75 or 1.35?

ehild