Two ice skaters, with masses of 40 kg and 85 kg , are at the center of a 50 m -diameter circular rink. The skaters push off against each other and glide to opposite edges of the rink. If the heavier skater reaches the edge in 30 s , how long does the lighter skater take to reach the edge?
momentum is conserved
... the two skaters have equal but opposite momenta
... M1 * V1 = M2 * V2
85 * (25 m / 30 s) = 40 * v
... solve for the lighter skater's velocity
... then find the time to travel 25 m
40/85 times the mas, so 85/40 times the speed.
Thus, 40/85 the time: 40/85 * 30 = ___ s
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum of the system remains constant before and after the skaters push off against each other.
The momentum of an object is given by the product of its mass and velocity.
Let's denote the mass of the heavier skater as M (85 kg) and the mass of the lighter skater as m (40 kg).
Initially, the skaters are at rest, so their initial momentum is zero. This means that the total momentum of the system after the push-off must also be zero.
Since the skaters are at opposite edges of the rink after the push-off, we can assume that the heavier skater moves in the positive direction (clockwise) and the lighter skater moves in the negative direction (counterclockwise).
Let's denote the velocity of the heavier skater as v1 and the velocity of the lighter skater as v2.
According to the conservation of momentum:
(M × v1) + (m × v2) = 0
Since the heavier skater reaches the edge of the rink in 30 seconds, we can use the distance formula:
v1 = d1 / t1
v1 = 50 m / 30 s
v1 = 5/3 m/s
Now we can substitute this value back into the momentum equation:
(85 kg × (5/3) m/s) + (40 kg × v2) = 0
Solving for v2, we find:
(40 kg × v2) = - (85 kg × (5/3) m/s)
v2 = - (85 kg × (5/3) m/s) / 40 kg
v2 = - 141.67/3 m/s
v2 = - 47.22 m/s
Since the magnitude of the velocity represents the speed, we take the positive value:
v2 = 47.22 m/s
Finally, we can use the distance formula again to find the time taken by the lighter skater to reach the edge:
v2 = d2 / t2
47.22 m/s = 50 m / t2
Solving for t2, we find:
t2 = (50 m) / (47.22 m/s)
t2 = 1.06 s
Therefore, the lighter skater takes approximately 1.06 seconds to reach the edge of the rink.
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the skaters push off against each other is equal to the total momentum after they reach the opposite edges of the rink.
The momentum of an object is calculated by multiplying its mass by its velocity. Let's denote the mass of the heavier skater as m1 = 85 kg and the mass of the lighter skater as m2 = 40 kg.
The total momentum before the skaters push off is zero since they are initially at rest. Therefore, the total momentum after they reach the opposite edges of the rink must also be zero.
Let's assume that the heavier skater moves with a velocity v1 and the lighter skater moves with a velocity v2. Consequently, the momentum of the heavier skater is given by p1 = m1 * v1, and the momentum of the lighter skater is p2 = m2 * v2.
Since the total momentum after the skaters reach the edges is zero, we have the equation p1 + p2 = 0.
Substituting the values for m1 and m2, we get (85 kg * v1) + (40 kg * v2) = 0.
However, we are asked to find the time it takes for the lighter skater to reach the edge. We can use the formula speed = distance / time to determine the velocity of each skater.
Since both skaters travel the same distance (the diameter of the circular rink), we can write the equation v1 * 30 s = v2 * t, where t is the time taken by the lighter skater to reach the edge.
Rearranging the equation, we get t = (v1 * 30 s) / v2.
To solve for t, we need to determine the ratio v1 / v2. From the equation (85 kg * v1) + (40 kg * v2) = 0, we can express v1 in terms of v2 as follows:
85 kg * v1 = -40 kg * v2
v1 = (-40 kg * v2) / 85 kg
v1 = (-8/17) * v2
Now we substitute this expression for v1 in the equation for time:
t = ((-8/17) * v2 * 30 s) / v2
t = (-240/17) s
Therefore, it takes the lighter skater approximately 14.12 seconds to reach the edge of the rink.