Two titanium spheres approach each other head-on with the same speed and collide elastically. After the collision, one of the spheres, whose mass is 300 g, remains at rest. What is the mass of the other sphere?

We will solve this problem step by step using the principles of elastic collisions and the conservation laws.

Step 1: Understand the Given Data
Two titanium spheres are moving towards each other with the same speed.
The collision is elastic (which means both momentum and kinetic energy are conserved).
After the collision, one of the spheres (mass = 300 g) comes to rest.
We need to find the mass of the other sphere.
Let:

Mass of the first sphere = m1 = 300 g = 0.300 kg
Mass of the second sphere = m2 (to be determined)
Initial speed of each sphere = v (same magnitude, opposite directions)
Final speed of the first sphere (m1) after collision = 0 m/s (since it comes to rest)
Final speed of the second sphere (m2) after collision = v2' (to be determined)
Step 2: Apply Conservation of Momentum
Since the collision is elastic, the total momentum before and after the collision must be the same.

Before Collision:
The spheres move towards each other with the same speed but in opposite directions, so the initial total momentum is:

Initial momentum = m1 * v + m2 * (-v) = m1 * v - m2 * v

After Collision:
The first sphere (m1) is at rest, so its momentum is zero, and the second sphere moves with speed v2':

Final momentum = m1 * 0 + m2 * v2' = m2 * v2'

Equating initial and final momentum:

m1 * v - m2 * v = m2 * v2' (Equation 1)

Step 3: Apply Conservation of Kinetic Energy
Since the collision is elastic, kinetic energy is also conserved.

Before Collision:
Total initial kinetic energy:

KE_initial = (1/2) * m1 * v^2 + (1/2) * m2 * v^2
= (1/2) * (m1 + m2) * v^2

After Collision:
Total final kinetic energy:

KE_final = (1/2) * m1 * 0^2 + (1/2) * m2 * (v2')^2
= (1/2) * m2 * (v2')^2

Since KE_initial = KE_final:

(1/2) * (m1 + m2) * v^2 = (1/2) * m2 * (v2')^2 (Equation 2)

Step 4: Solve for m2
From Equation 1:

m1 * v - m2 * v = m2 * v2'
=> v (m1 - m2) = m2 * v2'
=> m1 - m2 = m2 * (v2' / v) (Equation 3)

From Equation 2:

(m1 + m2) * v^2 = m2 * (v2')^2
=> (m1 + m2) * v^2 = m2 * v2'^2
=> v^2 = (m2 * v2'^2) / (m1 + m2)
=> v2'^2 = (m1 + m2) * v^2 / m2
=> v2' = v * sqrt((m1 + m2) / m2) (Equation 4)

Substituting Equation 4 into Equation 3:

m1 - m2 = m2 * sqrt((m1 + m2) / m2)
Squaring both sides:

(m1 - m2)^2 = m2 * (m1 + m2)
Expanding:

m1^2 - 2 * m1 * m2 + m2^2 = m2 * m1 + m2^2
Cancelling m2^2:

m1^2 - 2 * m1 * m2 = m2 * m1
=> m1^2 = 3 * m1 * m2
=> m2 = m1 / 3

Substituting m1 = 0.300 kg:

m2 = 0.300 / 3
m2 = 0.100 kg or 100 g

Step 5: Final Answer
The mass of the other sphere is 100 g or 0.100 kg.