(صورة توضيحية فقط)
The Galilean cannon demonstrates velocity multiplication through sequential, one-dimensional elastic collisions. When a stack of balls with decreasing mass is dropped, the bottom ball rebounds and collides with the one above. In an idealized case where a large mass [latex]m_1[/latex] rebounds at velocity [latex]v[/latex] and hits a much smaller mass [latex]m_2[/latex] moving down at [latex]v[/latex], the smaller mass is propelled upwards at nearly [latex]3v[/latex].
The core principle of the Galilean cannon relies on the conservation of linear momentum and kinetic energy in perfectly elastic collisions. Consider two balls, a large one of mass [latex]m_1[/latex] and a small one of mass [latex]m_2[/latex]. The entire stack falls under gravity, reaching a velocity [latex]-v[/latex] just before impact. The bottom ball, [latex]m_1[/latex], strikes the ground and perfectly rebounds with velocity [latex]+v[/latex]. It immediately collides with ball [latex]m_2[/latex], which is still moving downwards at [latex]-v[/latex].
From the perspective of an observer on ball [latex]m_1[/latex], ball [latex]m_2[/latex] is approaching at a relative velocity of [latex](-v) – (+v) = -2v[/latex]. In a perfectly elastic collision, the relative velocity of separation is equal to the negative of the relative velocity of approach. Therefore, after the collision, ball [latex]m_2[/latex] will be moving away from [latex]m_1[/latex] with a relative velocity of [latex]+2v[/latex].
To find the final velocity of [latex]m_2[/latex] in the lab frame, [latex]v_2′[/latex], we add this relative separation velocity to the final velocity of [latex]m_1[/latex], [latex]v_1′[/latex]. The formula for the final velocities in a 1D elastic collision gives [latex]v_2′ = \frac{v(3m_1 – m_2)}{m_1 + m_2}[/latex]. In the limiting case where [latex]m_1 >>>> m_2[/latex], the mass of [latex]m_1[/latex] is so large that its velocity is barely affected by the collision, so [latex]v_1′ \approx v[/latex]. The final velocity of [latex]m_2[/latex] is then [latex]v_2′ \approx v_1′ + 2v \approx v + 2v = 3v[/latex]. This tripling of velocity for the second ball is the fundamental amplification effect. If more balls are stacked, this effect cascades, leading to even higher velocities for the topmost ball.
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The Galilean Cannon – Velocity Multiplication in Stacked Ball Collisions
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الصور والتنزيلات بالحجم الكامل متاحة فقط، ومجانية تمامًا، للأعضاء المسجلين.
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