A 0.100 kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball.

(a).what is the mass of the second ball?
(b).What fraction of the original kinetic energy (ΔKE/KE) gets transferred to the second ball?

(a) Well, I could always check its driver's license, but I doubt the second ball has one. So, I'll have to rely on physics instead. Let's call the mass of the second ball "m2". Since these two balls are colliding with each other, according to the law of conservation of momentum, we have:

(m1 * v1)i = (m1 * v1)f + (m2 * v2)f

Where:
m1 = mass of the first ball (0.100 kg)
v1i = initial velocity of the first ball
v1f = final velocity of the first ball
v2f = final velocity of the second ball

The first ball is coming with an initial velocity, and the second ball is initially at rest. After the collision, the first ball's final velocity is half its initial velocity, which means v1f = (v1i)/2.

Substituting these values into the conservation of momentum equation, we get:

(0.100 kg * v1i) = (0.100 kg * (v1i)/2) + (m2 * v2f)

Simplifying this equation, we find:

v1i = v2f + (v1i)/2

Multiplying both sides by 2, we get:

2 * v1i = 2 * v2f + v1i

Rearranging the terms, we find:

v1i = 2 * v2f

So, the initial velocity of the first ball is twice the final velocity of the second ball. Now, since we know that m1 = 0.100 kg, we can solve for the mass of the second ball (m2):

(0.100 kg * 2 * v2f) = (m2 * v2f)

Dividing both sides by v2f, we're left with:

2 * 0.100 kg = m2

Therefore, the mass of the second ball is 0.200 kg.

(b) Now, let's calculate the fraction of the original kinetic energy that gets transferred to the second ball. The formula for kinetic energy is:

KE = (1/2) * m * v^2

Where:
m = mass
v = velocity

First, let's find the initial kinetic energy (KEi) of the first ball:

KEi = (1/2) * m1 * (v1i)^2

And the final kinetic energy (KEf) of the first ball:

KEf = (1/2) * m1 * (v1f)^2

The fraction of the original kinetic energy transferred to the second ball is given by:

ΔKE/KE = (KEf - KEi) / KEi

Substituting the values we know:

KEi = (1/2) * 0.100 kg * (v1i)^2
KEf = (1/2) * 0.100 kg * (v1f)^2

ΔKE/KE = ((1/2) * 0.100 kg * (v1f)^2 - (1/2) * 0.100 kg * (v1i)^2) / (1/2) * 0.100 kg * (v1i)^2

Simplifying this equation, we get:

ΔKE/KE = ((1/2) * (v1f)^2 - (1/2) * (v1i)^2) / (1/2) * (v1i)^2

Since v1f = (v1i)/2, we can substitute this value:

ΔKE/KE = ((1/2) * ((v1i)/2)^2 - (1/2) * (v1i)^2) / (1/2) * (v1i)^2

Simplifying further:

ΔKE/KE = ((1/2) * ((v1i)^2)/4 - (1/2) * (v1i)^2) / (1/2) * (v1i)^2

ΔKE/KE = (1/8) - 1

ΔKE/KE = -7/8

So, the fraction of the original kinetic energy transferred to the second ball is -7/8 (or approximately -0.875). Since this value is negative, it means that the second ball actually absorbs more energy than it initially had, which is pretty impressive for a little ball!

To solve this problem, we can use the principles of conservation of momentum and kinetic energy. Let's go step by step to find the answers:

(a) The conservation of momentum states that the total momentum before the collision is the same as the total momentum after the collision. Therefore, we can write:

(mass of first ball) x (initial velocity of first ball) = (mass of second ball) x (final velocity of second ball)

Given:
Mass of the first ball (m1) = 0.100 kg
Mass of the second ball (m2) = unknown
Initial velocity of the first ball (u1) = unknown
Final velocity of the second ball (v2) = (1/2) x (initial velocity of the first ball)

Using the principle of conservation of momentum, we can set up the equation:

(m1 x u1) = (m2 x v2)

Substituting the values, we have:

(0.100 kg) x (u1) = (m2) x [(1/2) x (u1)]

Simplifying the equation, we find:

0.100 kg = (1/2) x (m2)

To find the mass of the second ball (m2), we multiply both sides of the equation by 2:

0.200 kg = m2

Therefore, the mass of the second ball is 0.200 kg.

(b) The fraction of the original kinetic energy transferred to the second ball can be found using the conservation of kinetic energy, which states that the total kinetic energy before the collision equals the total kinetic energy after the collision. The formula for kinetic energy is:

KE = (1/2) x (mass) x (velocity^2)

Given:
Mass of the first ball (m1) = 0.100 kg
Mass of the second ball (m2) = 0.200 kg
Initial velocity of the first ball (u1) = unknown
Final velocity of the second ball (v2) = (1/2) x (initial velocity of the first ball)

The initial kinetic energy (KE1) can be calculated using the formula:

KE1 = (1/2) x (m1) x (u1^2)

The final kinetic energy (KE2) can be calculated by substituting the values into the formula:

KE2 = (1/2) x (m2) x (v2^2)

Using the equation:

KE1 = KE2

(1/2) x (m1) x (u1^2) = (1/2) x (m2) x (v2^2)

Substituting the given values, we have:

(1/2) x (0.100 kg) x (u1^2) = (1/2) x (0.200 kg) x [(1/2) x (u1)^2]^2

Simplifying the equation, we find:

0.100 x (u1^2) = 0.050 x (u1^2/4)

Multiplying both sides of the equation by 4, we get:

0.400 x (u1^2) = 0.050 x (u1^2)

Dividing both sides of the equation by (u1^2) and cancelling out the terms, we find:

0.400 = 0.050

Therefore, the fraction of the original kinetic energy transferred to the second ball is 0.050 or 5%.

To find the answers to these questions, we can use the principles of conservation of momentum and conservation of kinetic energy.

(a) To find the mass of the second ball (m2), we can start by using the conservation of momentum equation:

m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

Where:
m1 = mass of the first ball = 0.100 kg (given)
v1i = initial velocity of the first ball (before collision)
v1f = final velocity of the first ball (after collision) = 0.5 * v1i (given)
v2i = initial velocity of the second ball (before collision) = 0 m/s (given)
v2f = final velocity of the second ball (after collision)

Since the second ball is initially at rest (v2i = 0), the equation simplifies to:

m1 * v1i = m1 * v1f + m2 * v2f

Substituting the given values:

(0.100 kg) * v1i = (0.100 kg) * (0.5 * v1i) + m2 * v2f

Now we need to simplify the equation and solve for m2:

0.100 kg * v1i = 0.050 kg * v1i + m2 * v2f

0.100 kg * v1i - 0.050 kg * v1i = m2 * v2f

0.050 kg * v1i = m2 * v2f

Now, we need an additional piece of information to determine the value of v2f, which is the final velocity of the second ball after the collision. With that value, we can find the mass of the second ball.

(b) To find the fraction of the original kinetic energy (ΔKE/KE) transferred to the second ball, we can use the conservation of kinetic energy equation:

KE1i + KE2i = KE1f + KE2f

Where:
KE1i = initial kinetic energy of the first ball
KE2i = initial kinetic energy of the second ball (before collision) = 0 (since it is initially at rest)
KE1f = final kinetic energy of the first ball (after collision)
KE2f = final kinetic energy of the second ball (after collision)

The fraction of the original kinetic energy transferred to the second ball can be represented as ΔKE/KE.

ΔKE/KE = (KE2f - KE2i) / KE1i

Since the second ball is initially at rest (KE2i = 0), the equation simplifies to:

ΔKE/KE = KE2f / KE1i

Now, let's find the final kinetic energy of the second ball (KE2f). By substituting the initial and final velocities, we can calculate the kinetic energy:

KE2f = (1/2) * m2 * v2f^2

Once we find the values of v2f and KE2f, we can substitute them into the equation to determine the fraction of the original kinetic energy transferred to the second ball.

(a) Let m1 be the mass (0.1 kg) of the ball that is moving with velocity u1 before collision. Let its final velocity be v1. Let m2 be the unkown mass of the ball that is initially at rest. After collsion it has velocity v2.

Because of momentum conservation,
m1*u1 + m2u2 = m1v1 + m2v2
0.1 u1 + 0 = 0.1*v1 + m2*u1/2
u1 = v1 + 5m2*u1

Since the collision is elastic, the relative velocity of the two balls, before and after collsion, does not change magnitude, but changes direction.
v1 - v2 = u2 - u1
(You may need to derive this separately, or consider what happens in a coordinate system traveling with the center of mass. The relative velocity, which is independent of coordinate system, changes sign but keeps the same magnitude)

v1 - u1/2 = 0 - u1
v1 = -u1/2
Combining two previous equations,
u1 = -v1/2 + 5m2*u1

Therefore
5m*u1 = 3u1/2
m = 3/10 = 0.3 kg

(b) For the kinetic energy,
KE(m2),final = 1/2 * 0.3 * (u1/2)^2 = 0.0375 u1^2
and KE(m1),initial = 1/2 x 0.1 x u1^2 = 0.05u1^2
The ratio of KE gain of m2 to intial KE is
= ÄKE/KE = 0.0375u1^2/0.05u1^2 = 0.75