A 40-kg skater moving at 4 m/s overtakes a 60-kg skater moving at 2 m/s in the same direction and collides with her. The two skaters stick together. What is their final speed?
To find the final speed of the two skaters after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Mathematically, it can be expressed as:
Momentum = mass × velocity
Let's denote the mass and velocity of the first skater as m1 and v1, and the mass and velocity of the second skater as m2 and v2.
Before the collision, the momentum of the first skater is m1 × v1, and the momentum of the second skater is m2 × v2. Therefore, the total momentum before the collision is:
Total momentum before = (mass of the first skater × velocity of the first skater) + (mass of the second skater × velocity of the second skater)
Total momentum before = (m1 × v1) + (m2 × v2)
After the collision, the two skaters stick together and move as a single object with a combined mass (m1 + m2). Let's denote their final velocity as vf. The total momentum after the collision is:
Total momentum after = (mass of the combined skaters × final velocity after collision)
Total momentum after = (m1 + m2) × vf
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we can write:
(m1 × v1) + (m2 × v2) = (m1 + m2) × vf
Now we can substitute the given values into the equation to find the final velocity:
(m1 × 4) + (m2 × 2) = (m1 + m2) × vf
Substituting m1 = 40 kg, v1 = 4 m/s, m2 = 60 kg, and v2 = 2 m/s:
(40 × 4) + (60 × 2) = (40 + 60) × vf
160 + 120 = 100 × vf
280 = 100 × vf
Dividing both sides by 100:
2.8 = vf
Therefore, the final speed of the two skaters after the collision is 2.8 m/s.