Nothing in my physics book addresses this question. Can someone please help??
Two ice skaters, with masses of 69kg and 47kg , stand facing each other on a 19-m-wide frozen river. The skaters push off against each other, glide backward straight toward the river's edges, and reach the edges at exactly the same time.
How far did the 69kg skater glide?
69 v1 = 47 v2
so
v2 = (69/47)v1
19 = (v1+v2)t
19 (v1 + 1.47 v1) t
so
v1 t = distance of first = 19/2.47
= 7.69 meters
Well, it seems like those skaters really know how to create a perfectly synchronized ice ballet! Now, let's use some physics humor to solve this problem.
Since we're dealing with a conservation of momentum situation, we'll assume there are no external forces acting on the skaters. That means the initial momentum of the system should be equal to the final momentum.
Now, let me do my calculations with a sprinkle of humor! If the 69kg skater glided a distance of x meters, and the 47kg skater glided a distance of 19 - x meters (remember, they're gliding towards opposite edges), we can use the conservation of momentum to find the answer.
Here we go: 69kg * 0 m/s + 47kg * 0 m/s = 69kg * (0 m/s) + 47kg * (19 m/s - x m/s).
Now, let's simplify: 0 = 47kg * 19 m/s - 47kg * x m/s.
And, rearranging: 47kg * x m/s = 47kg * 19 m/s.
Finally, dividing by 47kg: x m/s = 19 m/s.
So, the 69kg skater glided a distance of 19 meters! They sure gave it their all to reach those edges at the same time. Keep gliding and laughing!
To solve this problem, we can make use of the principle of conservation of momentum and use the fact that the skaters reach the edges at the same time.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant, provided there is no external force acting on it. In this case, the only external force acting on the skaters is the friction between their skates and the ice, which is negligible.
Let's denote the initial velocities of the 69kg and 47kg skaters as v1 and v2, respectively. Since they push off against each other, their initial velocities are equal in magnitude but opposite in direction (v1 = -v2).
The total momentum before pushing off is given by:
Total momentum before = (Mass of skater 1 * Velocity of skater 1) + (Mass of skater 2 * Velocity of skater 2)
The total momentum after pushing off is zero because the skaters come to rest at the edges of the river. Therefore, we can set the total momentum before pushing off equal to zero:
(Mass of skater 1 * Velocity of skater 1) + (Mass of skater 2 * Velocity of skater 2) = 0
Substituting v1 = -v2, we can rewrite the equation as:
(Mass of skater 1 * -v2) + (Mass of skater 2 * v2) = 0
Simplifying the equation:
-69kg * v2 + 47kg * v2 = 0
-69kg * v2 + 47kg * v2 = 0
-22kg * v2 = 0
Dividing both sides by -22kg:
v2 = 0
This means that the velocity of the 47kg skater is zero, so the 69kg skater will continue gliding until reaching the edge on their own.
To find the distance traveled by the 69kg skater, we can use the equation for distance (d) traveled with constant velocity (v):
d = v * t
Since the time taken to reach the edges is the same for both skaters, we can use either skater's time. We'll assume the 47kg skater's time is t.
Given that the width of the river is 19m and the time is the same for both skaters, the distance traveled by the 69kg skater is:
d = v1 * t = v1 * (distance / v2)
Substituting the known values:
d = v1 * (19m / 0m/s) = undefined
Therefore, without the value of v1 (the initial velocity magnitude), we cannot determine the distance traveled by the 69kg skater.
To find out how far the 69kg skater glided, we can use the principle of conservation of momentum.
Let's assume that the initial velocity of the skaters is zero, meaning they were initially at rest before they pushed off against each other. According to the principle of conservation of momentum, the total momentum before and after the push-off must be the same.
The momentum of an object is given by the product of its mass and velocity. So, before the push-off, the total momentum of the system (skaters) is zero since their velocities are zero.
After the push-off, the skaters move in opposite directions, but they reach the edges at the same time. This implies that the force exerted by the skaters on each other is equal in magnitude and opposite in direction. This equal and opposite force causes the skaters to have equal but opposite velocities.
Let's assume that the 69kg skater moves a distance 'x' towards the river's edge. Since the 69kg skater glides backward, we can consider this distance as negative. The 47kg skater will then glide a distance '-x' towards the river's edge.
The principle of conservation of momentum can be expressed as:
(Initial momentum of the system) = (Final momentum of the system)
The initial momentum of the system is zero, and the final momentum is the sum of the individual momentums of the skaters. So, we have:
0 = (mass of the 69kg skater) x (velocity of the 69kg skater) + (mass of the 47kg skater) x (velocity of the 47kg skater)
Since the magnitudes of their velocities are equal, the equation becomes:
0 = 69kg x v - 47kg x v
Now we can solve for velocity 'v':
0 = (69kg - 47kg) x v
0 = 22kg x v
v = 0
This implies that the velocity of both skaters is zero when they reach the edges because they come to rest.
Now, since velocity is the change in distance over time, and the time taken for both skaters to reach the edges is the same, we can say that the distance traveled by the 69kg skater is equal to '-x'.
We have the equation:
Distance = Velocity x Time
Distance = 0 x Time
Distance = 0
So, according to the above analysis, the 69kg skater did not glide any distance.