Three uniform small spheres A, B and C have equal radii and masses 3 m, m and m respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with B between A and C. The coefficient of restitution between each pair of spheres is e. Sphere A is projected directly towards B
with speed u.
(i) Find, in terms of u and e, expressions for the speeds of A, B and C after the first two collisions
(ii)Given that A and C are moving with equal speeds after these two collisions, find the value of e
if x varies inversely as p^2 and x=2 when p=4, find the formulae connecting x and p.
i was thinking this but i'm not sure.
x/p^2
2/4^2
To find the expressions for the speeds of A, B, and C after the first two collisions, we can use the conservation of momentum and the coefficient of restitution.
(i) After the first collision between spheres A and B, the conservation of momentum in the horizontal direction gives us:
m (initial velocity of A) = 3m (final velocity of A) + m (final velocity of B) ---(1)
After the collision between A and B, the coefficient of restitution determines the ratio of their final velocities, which can be written as:
(e = (final velocity of B - final velocity of A) / (initial velocity of A)) ---(2)
Rearranging equation (2), we get:
final velocity of B - final velocity of A = e * initial velocity of A
Substituting this into equation (1):
m (initial velocity of A) = 3m (final velocity of A) + m (e * initial velocity of A)
Simplifying the equation:
(m + 3m) final velocity of A = (1 + 3e) initial velocity of A
final velocity of A = [(1 + 3e) / 4] * initial velocity of A ---(3)
Similarly, the final velocity of B can be found:
final velocity of B = [(3 - e) / 4] * initial velocity of A ---(4)
Now, for the second collision between B and C, again applying the conservation of momentum in the horizontal direction:
m (final velocity of B) = m (final velocity of C)
Using equation (4):
[(3 - e) / 4] * initial velocity of A = final velocity of C
(ii) Given that A and C are moving with equal speeds after these two collisions, we can write:
final velocity of A = final velocity of C
Using equation (3):
[(1 + 3e) / 4] * initial velocity of A = final velocity of C
Now, substituting the previously found equation (4):
[(1 + 3e) / 4] * initial velocity of A = [(3 - e) / 4] * initial velocity of A
[(1 + 3e) / 4] = [(3 - e) / 4]
Simplifying the equation:
1 + 3e = 3 - e
4e = 2
e = 1/2
Therefore, the value of e is 1/2.