Two ice skaters, with masses of 30kg & 85 kg are at the center of a 30m diameter circular rink. The skaters push off against each other & glide to opposite edges of the rink. If the heavier skater reaches the edge in 40s, how long does the lighter skater take to reach the edge?
Let the lighter skater have mass M1 and velocity V1 after pushing off. For the heaver skater, use M2 and V2.
Because of conservation of momentum,
M1 V1 + M2 V2 = 0
V1/V2 = -M2/M1 = -2.833
(The minus sign just means they go in opposite drections)
They both travel equal distances to the end of the rink. The time to get there is inversely proportional to the velocity and proporitonal the mass.
T2/T1 = |V1/V2| = M2/M1 = 2.833
You know that T2 = 40 s
Solve for T1
To solve this problem, we can use the principle of conservation of momentum. When the two skaters push off against each other, the total momentum of the system remains constant.
The equation for momentum is given by:
momentum = mass × velocity
Let's denote the mass of the lighter skater as m1 (30 kg) and the mass of the heavier skater as m2 (85 kg). The initial momentum of the system is zero since both skaters are initially at rest.
When they push off against each other, the lighter skater moves in one direction with velocity v1, while the heavier skater moves in the opposite direction with velocity v2. Since the total momentum is conserved, we have:
m1 × v1 = m2 × v2
We know the diameter of the circular rink is 30 m, which means the radius is 15 m. The distance traveled by each skater to reach the edge of the rink is equivalent to the circumference of a circle with a radius of 15 m, which is:
distance = circumference = 2π × radius = 2π × 15 m = 30π m
Now, we can use the speed-time-distance relationship to find the time taken for each skater to reach the edge of the rink. The formula is:
time = distance / speed
For the lighter skater:
time1 = (30π m) / v1
For the heavier skater:
time2 = (30π m) / v2
We are given that the heavier skater takes 40 s to reach the edge of the rink, so we can substitute that information into the equation:
40 s = (30π m) / v2
Now we can solve for v2:
v2 = (30π m) / (40 s) = (3π m) / (4 s)
Using this value of v2, we can solve for the time taken by the lighter skater:
time1 = (30π m) / v1
To find v1, we substitute the known values of v2, m2, m1, and rearrange the equation:
v1 = (m2 × v2) / m1 = (85 kg) × ((3π m) / (4 s)) / (30 kg)
Simplifying:
v1 = (85π m/s) / 40
Finally, we can substitute v1 into the equation for time1:
time1 = (30π m) / ((85π m/s) / 40)
Simplifying:
time1 ≈ (1200/85) s ≈ 14.12 s
Therefore, the lighter skater takes approximately 14.12 seconds to reach the edge of the circular rink.