At the center of a 50 m diameter circular ice rink, a 75 kg skater traveling north at 2.5 m/s collides with and holds onto a 60 kg skater who had been heading west at 3.5 m/s.
How long will it take them to get to the edge of the rink.
Where will the reach it. Give your answer as an angle north of west.
part 2: take your calculated velocities, divide them, and inverse tangent
To solve this problem, we can use the law of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
First, let's calculate the initial momentum of each skater:
Momentum of the first skater (north):
M1 = mass1 * velocity1 = 75 kg * 2.5 m/s = 187.5 kg·m/s
Momentum of the second skater (west):
M2 = mass2 * velocity2 = 60 kg * (-3.5 m/s) = -210 kg·m/s (Note: we use a negative sign because the direction is west)
Next, let's find the total momentum before the collision:
Total momentum before collision = M1 + M2 = 187.5 kg·m/s + (-210 kg·m/s) = -22.5 kg·m/s
Now, let's find the velocity of the skaters after the collision using the conservation of momentum:
Total momentum after collision = 0 (since they come to rest)
Momentum of the first skater after collision = mass1 * velocity1
Momentum of the second skater after collision = mass2 * velocity2
Since the skaters come to rest, the total momentum after the collision is zero. Therefore:
mass1 * velocity1 + mass2 * velocity2 = 0
Substituting the values:
75 kg * velocity1 + 60 kg * velocity2 = 0
Now, we can solve for the velocity of the first skater:
velocity1 = -(mass2 * velocity2) / mass1
velocity1 = -(60 kg * (-3.5 m/s)) / 75 kg
velocity1 = 2.8 m/s
Now, we know that the final velocity of the first skater is 2.8 m/s. Since only this skater is moving north, the velocity of the second skater after the collision is 0 m/s.
Now, let's calculate the time it takes for the skaters to reach the edge of the rink. Assuming that the skaters move along the circumference of the rink, the distance is equal to the circumference of the circle.
Circumference of the rink = 2 * π * radius = 2 * π * (50 m / 2) = 2 * π * 25 m = 50π m
Since velocity is equal to distance divided by time, we can rearrange the formula to solve for time:
Time = Distance / Velocity = (50π m) / 2.8 m/s ≈ 56.548 seconds
Finally, to find the angle north of west where they reach the edge, we can use trigonometry. Since the skater is initially moving west, and the radius is along the x-axis, the angle can be found using the tangent function:
Angle north of west = arctan((50 m / 2) / (50π m)) = arctan(1 / π) ≈ 0.318 radians ≈ 18.261 degrees
Therefore, it will take approximately 56.548 seconds for the skaters to reach the edge of the rink, and they will reach it at an angle approximately 18.261 degrees north of west.
To find out how long it will take for the skaters to reach the edge of the rink, we need to determine the resulting velocity after the collision. Since the skaters hold onto each other, they will move together as one unit.
First, let's calculate the momentum of each skater before the collision. The momentum of an object is given by the product of its mass and velocity.
For the first skater:
Momentum = mass × velocity = (75 kg) × (2.5 m/s) = 187.5 kg·m/s
For the second skater:
Momentum = mass × velocity = (60 kg) × (3.5 m/s) = 210 kg·m/s
Since momentum is conserved in a collision, the total momentum before the collision should be equal to the total momentum after the collision. Thus, we can calculate the resulting velocity of the combined skaters.
Total momentum before collision = Total momentum after collision
(75 kg × 2.5 m/s) + (60 kg × 3.5 m/s) = (Combined mass) × (Resulting velocity)
(187.5 kg·m/s) + (210 kg·m/s) = (Combined mass) × (Resulting velocity)
397.5 kg·m/s = (Combined mass) × (Resulting velocity)
Now, let's find the combined mass of the skaters:
Combined mass = 75 kg + 60 kg = 135 kg
Therefore:
397.5 kg·m/s = (135 kg) × (Resulting velocity)
(Resulting velocity) = 397.5 kg·m/s / 135 kg = 2.94 m/s
Now that we have the resulting velocity, we can calculate the time it takes for the skaters to reach the edge of the rink. Since the rink is circular with a 50 m diameter, the radius is half of that, which is 25 m.
Using the formula for linear motion, time = distance / velocity, we can calculate the time:
Time = (Radius of rink) / (Resulting velocity)
Time = 25 m / 2.94 m/s = 8.5 s (approximately)
Finally, to determine the direction they will reach the edge, we can use trigonometry. The angle north of west can be given by the inverse tangent (arctan) of the ratio of the vertical displacement (north) to the horizontal displacement (west).
This can be calculated as follows:
Angle north of west = arctan(vertical displacement / horizontal displacement)
The vertical displacement can be determined using the resulting velocity and the time:
Vertical displacement = Resulting velocity × Time
Vertical displacement = 2.94 m/s × 8.5 s = 24.99 m (approximately)
Since the horizontal displacement is the radius of the rink, which is 25 m, the angle north of west can be calculated as:
Angle north of west = arctan(24.99 m / 25 m) ≈ 45 degrees north of west
Therefore, it will take approximately 8.5 seconds for the skaters to reach the edge of the rink, and they will reach it at an angle approximately 45 degrees north of west.