Question

A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. The ratio of velocities of the balls after collision is x : y, where x is _________

Hint:

In symmetric collisions where identical particles scatter at equal angles, their final speeds must be equal to conserve momentum in the transverse direction.

Answer 1

Correct Answer: 1

To solve this problem, we must understand the conservation of momentum and the geometry of elastic collisions. Two identical balls imply that mass, m, is equal. The initial momentum is purely due to the moving ball.

Momentum before collision: p_initial = mv_initial = m × 9 m/s.

Post-collision, both balls make a 30° angle with the original direction. By symmetry and conservation of linear momentum, both balls move with similar angular dispersion.

We define velocities of the balls after collision as v₁ and v₂. Since both angles are 30°, resolve into components:

v_x1 = x cos(30°) = x √3/2 and v_y1 = x sin(30°) = x /2, same for y-direction ball.

Due to symmetry in an elastic collision, both balls must emerge with complementary angles, conserving momentum in the x-direction:

mv_x1 + mv_x2 = mv_initial.

Therefore: m(x√3/2) + m(y√3/2) = m × 9.

Simplifying with zero m reliance gives:

(x + y)√3/2 = 9.

For the y-direction, although initially zero, symmetry ensures:

mv_y1 - mv_y2 = 0.

Thus: x/2 = y/2, leading to x = y.

Substituting back: 2x√3/2 = 9, x√3 = 9, hence x = 9/√3 = 3√3 ≈ 5.20.

Substituting, we find: x = y, thus x = y = 3√3.

The ratio x : y simplifies to 1:1.

Confirming, this falls in the provided range of 1, 1.

Thus, x is 1.