At the centre of a circular ice rink, a 87.0 kg ice skater moving 2.10 m/s north hits and grabs onto a 51.0 kg ice skater who had been traveling west at 5.35 m/s. If the skaters hold onto each other and the rink has a 26.0 m diameter, what is the amount of time for the skaters to reach the edge of the rink?

To calculate the amount of time for the skaters to reach the edge of the rink, we can use the principle of conservation of momentum.

The formula for momentum is given by:

momentum = mass * velocity

First, let's calculate the momentum of both skaters before they grab onto each other. For the 87.0 kg skater moving north at 2.10 m/s:

momentum1 = mass1 * velocity1
= 87.0 kg * 2.10 m/s
= 182.7 kg·m/s (North)

For the 51.0 kg skater moving west at 5.35 m/s:

momentum2 = mass2 * velocity2
= 51.0 kg * 5.35 m/s
= 273.85 kg·m/s (West)

Since momentum is a vector quantity, we need to consider the directions. In this case, we have momentum1 in the North direction and momentum2 in the West direction.

When the skaters grab onto each other, their velocities become the same and they move together. The total momentum of both skaters after grabbing onto each other is equal to the sum of their individual momenta.

Let's calculate the total momentum:

total momentum = momentum1 + momentum2

Now, let's put the momentum values in the proper direction:

total momentum = 182.7 kg·m/s (North) + 273.85 kg·m/s (West)

To find the direction and magnitude of the total momentum, we can use the Pythagorean theorem:

magnitude of total momentum = √(north component^2 + west component^2)

magnitude of total momentum = √(182.7^2 + 273.85^2)

magnitude of total momentum ≈ 331.48 kg·m/s

Now, let's find the resultant velocity:

resultant velocity = magnitude of total momentum / total mass

The total mass is the sum of the masses of both skaters:

total mass = mass1 + mass2
= 87.0 kg + 51.0 kg
= 138.0 kg

resultant velocity = 331.48 kg·m/s / 138.0 kg
≈ 2.40 m/s

Now that we have the resultant velocity, we can calculate the time it takes for the skaters to reach the edge of the rink. The formula for the circumference of a circle is given by:

circumference = π * diameter

circumference = π * 26.0 m
≈ 81.68 m

The time can be calculated using the equation:

time = distance / velocity

time = 81.68 m / 2.40 m/s
≈ 34.03 s

Therefore, it would take approximately 34.03 seconds for the skaters to reach the edge of the rink.

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