A 59.0-kg skater is traveling due east at a speed of 1.80 m/s. A 61.0-kg skater is moving due south at a speed of 6.20 m/s. They collide and hold on to each other after the collision, managing to move off at an angle south of east, with a speed of vf. Find (a) the angle and (b) the speed vf, assuming that friction can be ignored.
To find the angle and speed after the collision, you can use the principles of conservation of momentum and conservation of kinetic energy.
Let's break down the steps to find the answers:
Step 1: Calculate the initial momenta of both skaters.
The formula for momentum is given as p = mv, where p is the momentum, m is the mass, and v is the velocity.
For the skater moving due east:
Mass (m1) = 59.0 kg
Velocity (v1) = 1.80 m/s
Momentum (p1) = m1 * v1
For the skater moving due south:
Mass (m2) = 61.0 kg
Velocity (v2) = -6.20 m/s (negative because it's heading south)
Momentum (p2) = m2 * v2
Step 2: Calculate the total momentum before the collision.
Since momentum is a vector quantity, we can treat east as the positive direction and south as the negative direction.
Total momentum before the collision (ptotal) = p1 + p2
Step 3: Calculate the angle of the final velocity.
Let θ be the angle between the final velocity and the east direction.
Using the momentum conservation in the east direction, we have:
p1cos(0°) = ptotalcos(θ)
m1v1 = ptotalcos(θ)
Using the momentum conservation in the south direction, we have:
p2sin(-90°) = ptotalsin(θ)
m2v2 = ptotalsin(θ)
Dividing the second equation by the first equation, we get:
(m2v2) / (m1v1) = (ptotalsin(θ)) / (ptotalcos(θ))
(m2v2) / (m1v1) = tan(θ)
Now we can calculate the angle θ using the inverse tangent function:
θ = tan^(-1)((m2v2) / (m1v1))
Step 4: Calculate the magnitude of the final velocity.
We can use the principle of conservation of kinetic energy to solve for the final velocity.
The initial kinetic energy (K) is given by the sum of the kinetic energy of each skater:
Kinitial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
The final kinetic energy (Kfinal) is given by the kinetic energy of the skaters moving together:
Kfinal = (1/2) * (m1 + m2) * vf^2
Since kinetic energy is conserved, Kinitial = Kfinal:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = (1/2) * (m1 + m2) * vf^2
Now solve for the final velocity vf:
vf = sqrt((m1 * v1^2 + m2 * v2^2) / (m1 + m2))
Step 5: Substitute the given values and calculate the final answers.
Substitute the given values of m1, v1, m2, and v2 into the equations for θ and vf.
After the calculations, you'll have:
(a) The angle θ between the final velocity and the east direction.
(b) The speed vf of the skaters after the collision.