On a cold winter day, a penny (mass 2.50 g) and a nickel (mass 5.00 g) are lying on the smooth (frictionless) surface of a frozen lake. With your finger, you flick the penny toward the nickel with a speed of 2.50 m/s .

The coins collide elastically; calculate both their final speeds. What are the directions of their final velocities?

To solve this problem, we can use the principles of conservation of momentum and kinetic energy.

Step 1: Determine the initial momentum of the system.
The initial momentum of the system is the sum of the momenta of the penny and the nickel. Since both coins are at rest initially, their momenta will be zero.

Step 2: Calculate the final momentum of the system.
After the collision, the momentum of the system is still conserved. Therefore, the final momentum of the system will also be zero.

Step 3: Apply the conservation of momentum to find the final velocities of the coins.
Since the final momentum is zero, the sum of the final momenta of the two coins must also be zero:

m1 * v1 + m2 * v2 = 0

where:
m1 is the mass of the penny (2.50 g = 0.0025 kg)
v1 is the final velocity of the penny
m2 is the mass of the nickel (5.00 g = 0.005 kg)
v2 is the final velocity of the nickel

Substituting the values:

0.0025 kg * v1 + 0.005 kg * v2 = 0

Step 4: Apply the conservation of kinetic energy to find the final speeds.
Since the collision is elastic, the total kinetic energy before and after the collision will be conserved. We can write the equation:

(1/2) * m1 * (v1)^2 + (1/2) * m2 * (v2)^2 = (1/2) * m1 * (2.50 m/s)^2

(1/2) * 0.0025 kg * (v1)^2 + (1/2) * 0.005 kg * (v2)^2 = (1/2) * 0.0025 kg * (2.50 m/s)^2

0.00125 * (v1)^2 + 0.0025 * (v2)^2 = 0.00625

Step 5: Solve the system of equations simultaneously.
We can solve the two equations above simultaneously to find the final velocities of the penny and nickel.

0.0025 * v1 + 0.005 * v2 = 0 (equation 1)
0.00125 * (v1)^2 + 0.0025 * (v2)^2 = 0.00625 (equation 2)

Solve equation 1 for v1:
v1 = -2 * v2

Substitute the value of v1 in equation 2:

0.00125 * (2v2)^2 + 0.0025 * v2^2 = 0.00625
0.005 * v2^2 + 0.0025 * v2^2 = 0.00625
0.0075 * v2^2 = 0.00625
v2^2 = 0.00625 / 0.0075
v2^2 = 0.8333

Taking the square root of both sides:

v2 = √0.8333
v2 ≈ 0.9135 m/s

Substituting the value of v2 in equation 1:

v1 = -2 * 0.9135
v1 ≈ -1.827 m/s

Step 6: Determine the directions of the final velocities.
The negative sign for v1 indicates that the penny is moving in the opposite direction to the initial flick. The positive value for v2 indicates that the nickel is moving in the same direction as the initial flick.

Therefore, the final speeds of the penny and nickel are approximately 1.827 m/s and 0.9135 m/s, respectively. The penny moves in the opposite direction, while the nickel moves in the same direction as the initial flick.

To calculate the final speeds and directions of the coins after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy. Here are the steps to find the solution:

Step 1: Determine the initial velocities of the coins.
The penny is flicked towards the nickel with a speed of 2.50 m/s. Since it is not mentioned explicitly, we assume that the initial velocity of the nickel is zero.

Step 2: Calculate the total initial momentum.
The total initial momentum can be calculated by multiplying the mass of each coin by its respective initial velocity and summing them up.
Initial momentum = (mass of penny * velocity of penny) + (mass of nickel * velocity of nickel)

Given:
Mass of the penny (m1) = 2.50 g = 0.0025 kg
Mass of the nickel (m2) = 5.00 g = 0.0050 kg
Velocity of the penny (v1) = 2.50 m/s
Velocity of the nickel (v2) = 0 m/s

Initial momentum = (0.0025 kg * 2.50 m/s) + (0.0050 kg * 0 m/s)

Step 3: Apply conservation of momentum.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Total initial momentum = Total final momentum

The total final momentum can be calculated by multiplying the mass of each coin by its respective final velocity and summing them up.
Total final momentum = (mass of penny * final velocity of penny) + (mass of nickel * final velocity of nickel)

Step 4: Apply conservation of kinetic energy.
Conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object can be calculated using the formula:
Kinetic energy = (1/2) * mass * velocity^2

The total initial kinetic energy can be calculated by summing the kinetic energy of both coins.
Total initial kinetic energy = (1/2 * mass of penny * velocity of penny^2) + (1/2 * mass of nickel * velocity of nickel^2)

The total final kinetic energy can be calculated by summing the kinetic energy of both coins using their respective final velocities.

Step 5: Solve for the final velocities.
With the conditions of conservation of momentum and conservation of kinetic energy, we have two equations (momentum and kinetic energy) and two unknowns (final velocities of the penny and nickel). We can solve these equations simultaneously to find the final velocities.

Solving these equations will yield the final velocities and their respective directions.

initial I = .00250 * 2.5

=
final I = .00250 * Vp + .00500 * Vn

Initial Ke = final K
=(1/2)(.00250)(6.25)
=(1/2)(.00250)Vp^2 + (1/2) (.00500) Vn^2