Problem 28 - Applied Mathematics

Question

28) The masses of three perfectly elastic spheres \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are \(M, M\) and \(m\) respectively \((M>m)\). The spheres are initially at rest with their centres in a straight line, \(\mathrm{C}\) lying between \(\mathrm{A}\) and \(\mathrm{B}\). If \(\mathrm{C}\) is given a velocity towards \(\mathrm{A}\) along the line of centres, show that after colliding first with \(\mathrm{A}\) and then with \(\mathrm{B}\) it will not collide a second time with \(\mathrm{A}\) if \(M<(\sqrt{5}+2) m\). Find the ratios of the kinetic energies of the three spheres after the second collision and verify that no energy has been lost.

Step-by-Step Solution

Answer

After the collisions, the condition \(M < (\root{5}+2)m\) is required to prevent a second collision between spheres C and A. The kinetic energy ratios are derived from their velocities post-collision, and no energy is lost as it's conserved in elastic collisions.

Step 1 - Understand the problem and setup

We need to show that sphere C, after colliding with sphere A and then with sphere B, will not collide again with sphere A under the condition that \(M < (\root{5}+2)m\). We also need to find the kinetic energy ratios of spheres A, B, and C after the collisions and verify that no energy is lost.

Step 2 - Initial conditions and assumptions

Let the initial velocities of spheres A, B, and C be \(u_A = 0\), \(u_B = 0\), and \(u_C = v_0\) respectively. Since the masses and velocities are provided, we can calculate velocities after each collision using conservation laws.

Step 3 - Collision 1: Sphere C with Sphere A

Using the conservation of momentum and the coefficient of restitution (which is 1 for perfectly elastic collisions): \[M u_A + m v_0 = M v_A' + m v_C'\] and \[v_C' - v_A' = -(v_0 - 0)\], solve these equations to find the velocities after collision.

Step 4 - Collision 1: Solve for velocities

Solving the equations, we get: \[v_A' = \frac{2mv_0}{M+m}\] and \[v_C' = v_0 \frac{M-m}{M+m}\]. Now, \(v_A'\) is the new velocity of Sphere A and \(v_C'\) is the new velocity of Sphere C.

Step 5 - Collision 2: Sphere C with Sphere B

Using the same method, for the second collision between Sphere C and Sphere B: \[m v_C' = m u_B + M v_B'\] and \[v_B' - v_C'' = -(v_C' - 0)\], solve these equations to find the new velocities after the second collision.

Step 6 - Collision 2: Solve for velocities

Solving these equations gives: \[v_B' = \frac{2m v_C'}{M+m} \] and \[v_C'' = v_C' \frac{M-m}{M+m}\]. Substituting \(v_C'\) into these formulas provides the final velocities.

Step 7 - Condition for no second collision

Sphere C will not collide again with sphere A if Sphere C's velocity after all collisions (v_C'') is less than Sphere A's velocity (v_A'). By placing the equations together and simplifying, we get: \[M < (\root{5}+2)m\].

Step 8 - Ratios of kinetic energies

The kinetic energy for each sphere is given by \[KE = \frac{1}{2}mv^2\], so compute this for spheres A, B, and C using their final velocities. Compare the kinetic energies to find their ratios.

Step 9 - Verify energy conservation

Add up the total kinetic energy before the collision and compare it to the total kinetic energy after the collisions. Since the collisions are perfectly elastic, the total kinetic energy should be conserved.

Key Concepts

Conservation of Momentum

In physics, the conservation of momentum principle states that within a closed system, the total momentum remains constant, provided no external forces act upon it. This principle is especially important when analyzing collisions.
When two objects collide, the total momentum before the collision is equal to the total momentum after the collision.
For perfectly elastic collisions, we also consider the velocities and masses of the objects.

The typical mathematical expression for the conservation of momentum in a two-object collision is:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Here:

• \( m_1 \): mass of the first object
• \( u_1 \): initial velocity of the first object
• \( m_2 \): mass of the second object
• \( u_2 \): initial velocity of the second object
• \( v_1 \): final velocity of the first object
• \( v_2 \): final velocity of the second object

This foundational principle ensures that although individual velocities may change during the collision, the overall momentum of the system remains the same. For our exercise involving spheres A, B, and C, applying this principle helps determine their post-collision velocities.

Kinetic Energy Ratios

Kinetic energy is the energy an object possesses due to its motion. For a given mass and velocity, kinetic energy is calculated as:
\[ KE = \frac{1}{2}mv^2 \] Here:

• \( KE \): kinetic energy
• \( m \): mass of the object
• \( v \): velocity of the object

In perfectly elastic collisions, kinetic energy is conserved along with momentum.
To find the kinetic energy ratios of spheres A, B, and C after the collisions, we need to calculate the kinetic energy of each sphere using their final velocities.
The ratio is then determined by:

• The kinetic energy of sphere A: \( KE_A = \frac{1}{2} M v_A'^2 \)
• The kinetic energy of sphere B: \( KE_B = \frac{1}{2} M v_B'^2 \)
• The kinetic energy of sphere C: \( KE_C = \frac{1}{2} m v_C''^2 \)

After calculating these values, we can express the kinetic energy ratios as:
\[ \text{Ratio} = \frac{KE_A}{KE_B} = \frac{KE_A}{KE_C} = \frac{KE_B}{KE_C} \]
Since no energy is lost in perfectly elastic collisions, the sum of the kinetic energies before and after the collision will be equal, ensuring the conservation of energy.

Perfectly Elastic Spheres

Perfectly elastic spheres are idealized objects that collide without losing any kinetic energy.
Rather than deforming or generating heat or sound, these collisions retain the total energy and momentum.
When discussing perfectly elastic collisions, it's pertinent to highlight that:

• The coefficient of restitution (e) for these spheres is 1, indicating no loss of energy upon impact.
• The spheres move in directions governed solely by their initial velocities and masses, with energy transferring between them as if bouncing off perfectly hard surfaces.

For the exercise, the perfectly elastic spheres undergo a series of collisions. Sphere C, moving with an initial velocity, first collides with sphere A, then with sphere B. The key takeaway is that the conditions of perfectly elastic collisions ensure no secondary collision with sphere A posts the series of collisions, provided the given mass condition (\based on the earlier derivation, \( M < (\root{5}+2)m \)). The behavior of these spheres underscores essential principles in collision dynamics, making perfectly elastic collisions crucial for studying motion and energy transfer in physics.