3. For a 2-body spinless elastic collision between equal masses (such as when pl

Question

3. For a 2-body spinless elastic collision between equal masses (such as when playing "pool" or "billiards"-but ignoring spin or "english" here), with one of the particles originally at rest, say the particles collide and fly off at some angle as shownm below. Use momentum and kinetic energy conservation to derive the total angle the balls fly off at (0, + 2 in the picture). HINT: Set up the momentum and kinetic energy conservation equations in two dimensions and then combine the two momentum equations to look like the kinetic energy equation-then the "math cancellation magic" happens and you have a beautiful solution Initial State y at rest m2X V. Final State y V.

Explanation / Answer

along y axis

piy = 0

Pfy = m1*v1*sinteta - m2*v2*sintheta2

Pfy = Piy

m1*v1*sinteta1 - m2*v2*sintheta2 =

m1 = m2

v1*sintheta1 = v2*sintheta2

v2 = v1*sintheta1/sintheta2................(1)

initial momentum Pix = m1*u1

after collision

along x axis

final momentum Pfx = m1*v1*costheta + m2*v2*costheta2

Pfx = Pix

m1*v1*costheta1 + m2*v2*costheta2 = m1*u1

m1 = m2

v1*costheta1 + v2*costheta2 = u1..............(2)

using 1 in 2

v1*costheta1 + v1*(sintheta1/sintheta2)*costheta2 = u1

v1*sintheta2*costheta1 + v1*sintheta1*costheta2 = u1*sintheta2

v1*sin(theta1 + theta2) = u1*sintheta2

squaring on both sides

v1^2(sintheta1+sintheta2)^2 = u1^2*(sintheta2)^2 ................(3)

from energy conservation

KEf = KEi

(1/2)*m1*v1^2 + (1/2)*m2*v2^2 = (1/2)*m1*u1^2

m1 = m2

v1^2 + v2^2 = u1^2

v2 = v1*(sintheta1/sintheta2)

v1^2 + v1^2*(sintheta1/sintheta2)^2 = u1^2

v1^2*( (sintheta1)^2 + (sintheta2)^2 ) = u1^2*(sintheta2)^2......(4)

3 = 4

(sin(theta1+theta2))^2 = ( (sintheta1)^2 + (sintheta2)^2 )

(sintheta1*costheta2)^2 + (costheta1*sintheta2)^2 + 2*(sintheta1*sintheta2*costheta1*costheta2) = ( (sintheta1)^2 + (sintheta2)^2 )

(sintheta1)^2 *(costheta2)^2 + ((costheta1)^2*(sintheta2)^2) + 2*(sintheta1*sintheta2*costheta1*costheta2) = ( (sintheta1)^2 + (sintheta2)^2 )

(sintheta1)^2*(1-(costheta2)^2) + (sintheta2)^2*( 1 - (costheta1)^2 ) = 2*(sintheta1*sintheta2*costheta1*costheta2)

(sintheta1)^2*(sintheta2)^2 + (sintheta2)^2*(sintheta1)^2 = 2*(sintheta1*sintheta2*costheta1*costheta2)

2*(sintheta1)^2*(sintheta2)^2 = 2*(sintheta1*sintheta2*costheta1*costheta2)

sintheta1*sintheta2 = costheta1*costheta2

tantheta1*tantheta2 = 1

tantheta1*tantheta2 = tantheta*tan(90-theta)

theta1 = theta

theta2 = 90 - theta

theta1 + theta2 = 90 <<<---------------ANSWER

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