Question:
A cart with a toy projectile launcher attached to its top travels forward at a constant speed vo. The launcher fires a solid sphere forward at speed much greater than that of the cart-launcher system. The cart's speed after firing the dart is vf <vo. Frictional forces are considered to be negligible. Which of the following scenarios best represents the rotational analog to this situation?
(A) A gaseous planet rotates with a large rotational velocity, causing the middle of the planet's equator to bulge outward.
(B) A spinning platform falls onto a stationary platform. Both platforms rotate together, but at a smaller common rotational speed than that of the spinning platform immediately before the collision
(C) A child on the edge of a large rotating disk throws a ball in the same direction as the child's tangential velocity
(D) A figure skater spins with the skater's hands at the skater's waist. When the skater extends the skater's arms outward, the skater's rotational velocity decreases.
The correct option that best represents an analog of the cart with a toy projectile launcher is;
(C) A child on the edge of a large rotating disk throws a ball in the same direction as the child's tangential velocity.
Explanation:
Here we have
Initial momentum = v₀×(m₁ + m₂ + m₃)
Where:
m₁ = Mass of cart
m₂ = Mass of the launcher
m₃ = Mass solid sphere
Therefore, by the principle of conservation of linear momentum, we have;
Sum of the initial and final momentum of the system is zero. That is;
Initial momentum of system = Final momentum of the system
v₀×(m₁ + m₂ + m₃) = 
× (m₁ + m₂) + v₂·m₃
Where:
v₂ = Speed of the sphere after it is fired
We are going to analyze each of the options to find the rotational analog.
(A) A gaseous planet rotates with a large rotational velocity, causing the middle of the planet's equator to bulge outward.
Here, we have a single mass system rotating causing the loosely bound components to assume a different position, bulge, to maintain equilibrium.
Therefore, there is only one velocity in the system, hence it is not an analog to the system in question.
(B) A spinning platform falls onto a stationary platform. Both platforms rotate together, but at a smaller common rotational speed than that of the spinning platform immediately before the collision
Here, we have, the initial velocity of the stationary platform is zero therefore it is not a perfect analog, however, it satisfies the reverse of the process in question to a point, except for the condition of the stationary platform.
(C) A child on the edge of a large rotating disk throws a ball in the same direction as the child's tangential velocity
Here, the child and the ball where both having an equal initial rotational velocity, which satisfies the first condition,
The child then throws a ball forward in the child's tangential velocity. This is analogous to the firing of the sphere.
This system is analogous to the system in question.
(D) A figure skater spins with the skater's hands at the skater's waist. When the skater extends the skater's arms outward, the skater's rotational velocity decreases.
Here the component of the system all remain together before and after the momentum change, hence, it is not an analog
