Help solve all this question please

[tauwee]

help solve all this question please...:(

A uniform smooth sphere P,of mass 3m, is moving in a straight line with speed u on a smooth horizontal table.Another uniform smooth sphere Q, of mass and m and having the same radius as P, moving with speed 2u in the same straight line as P but in the opposite direction to P. The sphere P collides with the sphere Q directly. The velocities of P and Q after the collision are v and w respectively, measured in the direction of motion of P before the collision. The coefficient of restitution between P and Q is e.

a) Find an expression for v in terms of u and e.
b) Show that, if the direction of motion and P is changed by the collision, then e>1/3.
c) Find an expression for w in terms of u and e.

Following the collision with P, the sphere Q then collides with and rebounds from a vertical wall which is perpendicular to the direction of motion of Q. The coefficient of restitution between Q and the wall is e'

Given that e=5/9, and that P and Q collide again in the subsequent motion,
e) Show that e'>1/9.

 

[sjb-2812]

Do you know any equations that may help?

 

[tauwee]

this is the prove question...i don't know wht equation can solve :(

 

[sjb-2812]

Can you define " The coefficient of restitution"?

 

[Nickie]

a) Using the conservation of momentum and the coefficient of restitution, we can set up the following equations:

Conservation of momentum: 3mu + m(-2u) = 3mv + mw
Coefficient of restitution: (v-w)/(u-(-2u)) = 5/9

Solving for v, we get: v = (10/9)u

b) If the direction of motion of P is changed by the collision, then v will be in the opposite direction as u. This means that v will be negative. Substituting this into the equation for the coefficient of restitution, we get:

(-|v|-w)/(-|u|-(-2u)) = 5/9
Simplifying, we get: (v-w)/(3u) = 5/9
Substituting the value of v from part a), we get: (-10/9u - w)/(3u) = 5/9
Simplifying, we get: w = (-5/3)u

Since w is negative, this means that the direction of motion of Q is also changed by the collision. This also means that the coefficient of restitution must be greater than 1/3, as the negative sign indicates a reversal in direction.

c) Using the conservation of momentum and the coefficient of restitution, we can set up the following equations:

Conservation of momentum: 3mu + m(-2u) = 3mv + mw
Coefficient of restitution: (v-w)/(u-(-2u)) = 5/9

Solving for w, we get: w = (-5/3)u

e) Using the conservation of momentum and the coefficient of restitution, we can set up the following equations:

Conservation of momentum: 3mv + mw = 3mv' + m(-2v')
Coefficient of restitution: (v'-(-2v'))/(w-(-2v')) = e'

Substituting the values for v and w from part a) and c), we get:

(10/9u - (-4/3)u)/(5/3u - (-4/3)u) = e'
Simplifying, we get: (28/27) = e'

Since e' is greater than 1/3, this means that e' must also be greater than 1/9, as shown.