A 51.5-kg skater is traveling due east at a speed of 2.70 m/s. A 69.5-kg skater is moving due south at a speed of 7.50 m/s. They collide and hold on to each other after the collision, managing to move off at an angle θ south of east, with a speed of vf. Find the following.
initial momentum=final momentum
51.5*2.7E+69.5*7.5=(51.5+69.5)V
so there are a number of ways to do this, because of the right angles,
(51.5+69.5)V= sqrt((51.5*2.7)^2+(69.5*7.5)^2)
solve for V,
it is at an angle theta measured E of S as
arctan((51.5*2.7)/(69.5*7.5) )
To find the desired quantities, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
1) Find the final velocity (vf) of the skaters after the collision:
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v).
Before the collision:
Momentum of Skater 1 (p1) = Mass of Skater 1 (m1) × Velocity of Skater 1 (v1)
p1 = (51.5 kg) × (2.70 m/s) = 139.05 kg·m/s (eastward)
Momentum of Skater 2 (p2) = Mass of Skater 2 (m2) × Velocity of Skater 2 (v2)
p2 = (69.5 kg) × (-7.50 m/s) = -521.25 kg·m/s (southward) [Note: The direction is southward, so the velocity is negative.]
Total initial momentum = p1 + p2 = 139.05 kg·m/s (eastward) - 521.25 kg·m/s (southward)
After the collision:
Total final momentum = (51.5 kg + 69.5 kg) × vf (vf is the final velocity of both skaters combined)
Using the principle of conservation of momentum:
Total initial momentum = Total final momentum
139.05 kg·m/s (eastward) - 521.25 kg·m/s (southward) = (121 kg) × vf (eastward component of momentum - southward component of momentum)
To find the final velocity (vf), we need to break it down into its eastward and southward components. Given that θ is the angle south of east, we can express the eastward component of vf as vf · cos(θ) and the southward component as vf · sin(θ).
Therefore, the equation becomes:
139.05 kg·m/s (eastward) - 521.25 kg·m/s (southward) = (121 kg) × vf · cos(θ) - (121 kg) × vf · sin(θ)
2) Find the angle θ:
To find the angle θ, we can use the equation from step 1 and solve it for θ.
Rearrange the equation:
139.05 kg·m/s - 521.25 kg·m/s = 121 kg × vf · cos(θ) - 121 kg × vf · sin(θ)
Combine like terms:
-382.20 kg·m/s = 121 kg × vf · cos(θ) - 121 kg × vf · sin(θ)
Rearrange the terms:
121 kg × vf · sin(θ) - 121 kg × vf · cos(θ) = -382.20 kg·m/s
Divide both sides by 121 kg × vf:
sin(θ) - cos(θ) = -382.20 kg·m/s / (121 kg × vf)
Finally, use trigonometric identities to simplify the equation and solve for θ.
3) Find the magnitude of the final velocity (vf):
Once you have the angle θ, you can calculate the magnitude of the final velocity (vf) by substituting its value back into the equation from step 1.
Total final momentum = (121 kg) × vf
Solve for vf.
Using these steps, you can find the desired quantities in the given scenario.