Skater N, 58.9 kg, is moving north at a speed of 7.8 m/s when she collides with Skater E, 72.6 kg, moving east at 3.5 m/s. The two skaters are stuck together. In what direction and with what speed do they move after the collision?
To determine the direction and speed at which Skater N and Skater E move after the collision, we can apply the principles of conservation of momentum.
1. Calculate the initial momentum of Skater N (PN_initial) using the formula:
PN_initial = mass_N * velocity_N
PN_initial = 58.9 kg * 7.8 m/s
2. Calculate the initial momentum of Skater E (PE_initial) using the same formula:
PE_initial = mass_E * velocity_E
PE_initial = 72.6 kg * 3.5 m/s
3. Express the initial momentum of each skater in vector form:
PN_initial = PN_initial, North
PE_initial = PE_initial, East
4. Since momentum is conserved, the total initial momentum before the collision should equal the total final momentum after the collision. Therefore:
PN_initial + PE_initial = PN_final + PE_final
5. The direction of the total final momentum can be determined using the principle of vector addition. Since Skater N was initially moving north and Skater E was initially moving east, the total final momentum will form a right triangle. The direction of the total final momentum can be found using the tangent of the angle:
tan(θ) = opposite/adjacent
θ = atan(opposite/adjacent)
6. Calculate the magnitude of the total initial momentum using Pythagorean's theorem:
magnitude_initial = sqrt((PN_initial^2) + (PE_initial^2))
7. Similarly, calculate the magnitude of the total final momentum using Pythagorean's theorem:
magnitude_final = sqrt((PN_final^2) + (PE_final^2))
8. Apply conservation of momentum to find the values of PN_final and PE_final.
By following this procedure, you can find the direction and magnitude of the combined momentum of Skater N and Skater E after the collision.
To determine the direction and speed of the skaters after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
First, let's calculate the momentum of each skater before the collision:
Momentum of Skater N before the collision = mass of Skater N x velocity of Skater N
= 58.9 kg x 7.8 m/s
= 459.42 kg·m/s (north direction)
Momentum of Skater E before the collision = mass of Skater E x velocity of Skater E
= 72.6 kg x 3.5 m/s
= 253.5 kg·m/s (east direction)
Now, let's calculate the total momentum before the collision:
Total momentum before the collision = Momentum of Skater N + Momentum of Skater E
= 459.42 kg·m/s (north direction) + 253.5 kg·m/s (east direction)
We can combine the north and east components of the momenta using vector addition. Since the momenta are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the resultant momentum:
Magnitude of resultant momentum = √(momentum of Skater N)^2 + (momentum of Skater E)^2
= √((459.42 kg·m/s)^2 + (253.5 kg·m/s)^2)
≈ 518.38 kg·m/s
The direction of the resultant momentum can be found using trigonometry. We can use the inverse tangent function to find the angle:
Angle = arctan((momentum of Skater N) / (momentum of Skater E))
= arctan((459.42 kg·m/s) / (253.5 kg·m/s))
≈ 61.31° (north of east)
So, after the collision, the skaters move at a speed of approximately 518.38 kg·m/s in a direction approximately 61.31° north of east.