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 calculated by multiplying its mass by its velocity. Let's denote the mass of the first skater as m1 (40 kg) and their initial velocity as v1 (4 m/s). Similarly, let's denote the mass of the second skater as m2 (60 kg) and their initial velocity as v2 (2 m/s).
The total momentum before the collision is given by:
Initial momentum = m1 * v1 + m2 * v2
After the collision, the two skaters stick together and move with the same final velocity, which we need to find. Let's represent this final velocity as v_f.
Now, since the two skaters stick together, their combined mass is the sum of their individual masses:
Total mass = m1 + m2
Therefore, the momentum after the collision is:
Final momentum = Total mass * v_f
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
m1 * v1 + m2 * v2 = (m1 + m2) * v_f
Substituting the given values:
(40 kg * 4 m/s) + (60 kg * 2 m/s) = (40 kg + 60 kg) * v_f
Simplifying further:
160 kg * m/s + 120 kg * m/s = 100 kg * v_f
280 kg * m/s = 100 kg * v_f
Dividing both sides by 100 kg gives:
2.8 m/s = v_f
Hence, the final speed of the two skaters after the collision is 2.8 m/s.