Two people, George who has a mass of 93.0 kg and Victoria who has a mass of 60.0 kg, are on a frozen circular pond. They stand at the very centre and push off of each other. The distance from one side of the pond to the other is 83.0 m and it takes George 15.0 s to reach the edge of the pond. How long does it take Victoria to reach the edge of the pond?
get speeds from conservation of momentum
93 g = 60 v
15 g + v t = 83
15 g = 83/2
---------------
so
--------------
g = 83/30 = 2.77
v = 93/60 *83/30
= 4.29
v t = 41.5
t = 9.67 seconds
check
9.67 * 4.29 = 41.5
15 * 2.77 = 41.55
v/g = 4.29/2.77 = 1.55
93/60 = 1.55 ok
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before and after the interaction should be the same.
1. First, let's determine the initial momentum of the system. Since both George and Victoria are standing still before they push off each other, their initial momentum is 0 kg•m/s.
2. Next, we'll determine the final momentum of the system. After George reaches the edge of the pond, he stops, so his final momentum is also 0 kg•m/s.
3. We can use the principle of conservation of momentum to find Victoria's final momentum. Since the total momentum before and after the interaction should be the same, we have:
Initial momentum (George) = Final momentum (Victoria)
0 kg•m/s = mass (Victoria) × velocity (Victoria)
4. We can rearrange the equation to solve for the velocity of Victoria:
velocity (Victoria) = 0 kg•m/s / mass (Victoria)
5. Now we need to find the distance that Victoria needs to travel to reach the edge of the pond. Since they both start from the same point at the center of the pond, the distance Victoria needs to travel is equal to the diameter of the pond, which is 83.0 m.
6. Finally, we can use the formula for velocity to find the time it takes Victoria to reach the edge of the pond:
time = distance / velocity
Now we can substitute the given values into the equations to find the answer.
Mass of Victoria = 60.0 kg
Velocity of Victoria = 0 kg•m/s / 60.0 kg = 0 m/s (since her final momentum is also 0 kg•m/s)
Distance = 83.0 m
Time = 83.0 m / 0 m/s
Time = NaN (Not a Number)
Since the velocity of Victoria is 0 m/s, it will take an infinite amount of time for her to reach the edge of the pond.
To solve this problem, we can use the conservation of momentum principle. Since George and Victoria push off each other, their total momentum remains constant before and after the push.
The formula for momentum is:
momentum = mass x velocity
Let's denote George's mass as m1 and Victoria's mass as m2. Their respective velocities before the push can be denoted as v1 and v2.
Since they are initially at rest (zero velocity), the total initial momentum is zero:
initial momentum = (m1 x v1) + (m2 x v2) = 0
After the push, George reaches the edge of the pond, so his velocity can be calculated using the formula:
distance = velocity x time
v1 = distance / time
v1 = 83.0 m / 15.0 s = 5.53 m/s
Now, we can calculate the initial momentum:
initial momentum = (93.0 kg x 5.53 m/s) + (60.0 kg x 0 m/s) = 514.29 kg·m/s
Since momentum is conserved, the final momentum is equal to the initial momentum:
final momentum = (m1 x v1') + (m2 x v2')
where v1' and v2' are the velocities of George and Victoria after the push.
Since Victoria starts from the center of the pond and George has reached the edge, their velocities are in opposite directions and have equal magnitudes:
v1' = -v2'
Hence, we can rewrite the final momentum equation as:
final momentum = (m1 x v1') - (m2 x v1') = (m1 - m2) x v1'
Plugging in the values, we have:
514.29 kg·m/s = (93.0 kg - 60.0 kg) x v1'
Simplifying, we find:
514.29 kg·m/s = 33.0 kg x v1'
Dividing both sides by 33.0 kg:
v1' = 15.59 m/s
Now, we can use the formula:
time = distance / velocity
to find the time it takes for Victoria to reach the edge of the pond:
time = 83.0 m / 15.59 m/s ≈ 5.33 s
Therefore, it takes Victoria approximately 5.33 seconds to reach the edge of the pond.