Two titanium spheres approach each other hand-on with the same speed and

collide elastically. After the collision, one of the shares, whose mass is 300 g, remains at

rest. What is the mass of other sphere?

To solve the problem of two titanium spheres colliding elastically, we need to apply the principles of conservation of momentum and conservation of kinetic energy. Let's break this down step by step.

Understanding Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum before the collision equals the total momentum after the collision, and the same goes for kinetic energy.

Given Information

• Mass of the first sphere (m1) = 300 g = 0.3 kg
• Mass of the second sphere (m2) = ? (this is what we need to find)
• Both spheres approach each other with the same speed (let's denote this speed as v).
• After the collision, one sphere (m1) remains at rest.

Applying Conservation of Momentum

The formula for conservation of momentum before and after the collision can be expressed as:

Initial Momentum = Final Momentum

Before the collision, the momentum of the system is:

m1 * v + m2 * (-v) = 0.3 kg * v - m2 * v

After the collision, since m1 is at rest, the momentum is:

0 + m2 * v' (where v' is the velocity of m2 after the collision)

Setting these equal gives us:

0.3 kg * v - m2 * v = m2 * v'

Applying Conservation of Kinetic Energy

For kinetic energy, the equation is:

Initial Kinetic Energy = Final Kinetic Energy

Before the collision, the total kinetic energy is:

KE_initial = 0.5 * m1 * v^2 + 0.5 * m2 * v^2

After the collision, since m1 is at rest, the kinetic energy is:

KE_final = 0.5 * m2 * (v')^2

Setting these equal gives us:

0.5 * 0.3 kg * v^2 + 0.5 * m2 * v^2 = 0.5 * m2 * (v')^2

Finding the Mass of the Second Sphere

Since m1 comes to rest after the collision, we can deduce that m2 must be equal to m1 for the collision to be elastic and for one sphere to stop completely. This is because, in an elastic collision where one object comes to rest, the other object must have a mass that is equal to the first object.

Thus, the mass of the second sphere (m2) is:

m2 = m1 = 300 g = 0.3 kg

Final Thoughts

In summary, both titanium spheres have the same mass of 300 grams. This outcome is a fascinating illustration of how conservation laws govern the behavior of objects in motion, particularly in elastic collisions. If you have any more questions about this topic or related concepts, feel free to ask!