#1
A smooth groove in the form of a circle of radius a is carved out of a horizontal table. Two small spheres, A and B, lie at rest in the groove at opposite ends of a diameter. At time t = 0, the sphere A is projected along the groove and the first collision occurs at time = T. Given that e is the coefficient of restitution between the spheres, find the velocities of A and B after the first collision. Hence, or otherwise, show that the second collision takes place at time

t = T(2+e)/e.

and
https://nrich.maths.org/discus/messa...tml?1172521717

on the same question however I cannot understand their explanations and are incomplete.

I cannot even obtain the velocities I have so many unknown. Grrr! Do I do the conservation of momentum thing and law of restitution? I have 5 unknowns!

0
9 years ago
#2
(Original post by lilangel890)
A smooth groove in the form of a circle of radius a is carved out of a horizontal table. Two small spheres, A and B, lie at rest in the groove at opposite ends of a diameter. At time t = 0, the sphere A is projected along the groove and the first collision occurs at time = T. Given that e is the coefficient of restitution between the spheres, find the velocities of A and B after the first collision. Hence, or otherwise, show that the second collision takes place at time

t = T(2+e)/e.

and
https://nrich.maths.org/discus/messa...tml?1172521717

on the same question however I cannot understand their explanations and are incomplete.

I cannot even obtain the velocities I have so many unknown. Grrr! Do I do the conservation of momentum thing and law of restitution? I have 5 unknowns!

initial velocity of A = V
velocity of A after impact = Va
velocity of B after impact = Vb
coefficient of restitution = e

Conservation of momentum states (cancelling masses as they are all equal) :-
V = Va + Vb

1) Va = V - Vb

Coefficient of restitution:
e = (Vb-Va)/V

=> eV = Vb-Va

Substituting 1)

=> eV = Vb-V+Vb
eV+V = 2Vb

2) Vb = (eV+V)/2

Substituting back in to 1

3) Va = V - ((eV+V)/2)

First sphere travelled semi-circle in time T, before collision so

4) V = (pi.a)/T

Substituting 4) into 2)

Vb = (e(pi.a/T)+(pi.a/T))/2

=> 2Vb = (e.pi.a + pi.a)/T

=> Vb = (e.pi.a + pi.a)/2T

5) so, Vb = (pi.a(e+1))/2T

and substituting 5) and 4) into 1) we can find Va

Va = (pi.a(1-e))/2T
0
#3
(Original post by rbnphlp)
initial velocity of A = V
velocity of A after impact = Va
velocity of B after impact = Vb
coefficient of restitution = e

Conservation of momentum states (cancelling masses as they are all equal) :-
V = Va + Vb

1) Va = V - Vb

Coefficient of restitution:
e = (Vb-Va)/V

=> eV = Vb-Va

Substituting 1)

=> eV = Vb-V+Vb
eV+V = 2Vb

2) Vb = (eV+V)/2

Substituting back in to 1

3) Va = V - ((eV+V)/2)

First sphere travelled semi-circle in time T, before collision so

4) V = (pi.a)/T

Substituting 4) into 2)

Vb = (e(pi.a/T)+(pi.a/T))/2

=> 2Vb = (e.pi.a + pi.a)/T

=> Vb = (e.pi.a + pi.a)/2T

5) so, Vb = (pi.a(e+1))/2T

and substituting 5) and 4) into 1) we can find Va

Va = (pi.a(1-e))/2T
Yeah, nice copy and paste there. I did say I didn't really understand that I have the velocities now but the 2nd part is confusing
0
9 years ago
#4
(Original post by lilangel890)
Yeah, nice copy and paste there. I did say I didn't really understand that I have the velocities now but the 2nd part is confusing
..You mean show that
0
#5
(Original post by rbnphlp)
..You mean show that
What?
0
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