Two skaters, Alan and Betsy, move toward each other on a very smooth skating rink. Their masses and speeds are respectively: 60 kg; 4.0 m/s and 50 kg; 0.4 m/s. After the skaters collide and use their hands to push each other away, Alan is moving backwards at 2.0 m/s. Taking the direction of Alan's initial velocity as positive, find Betsy's velocity vf after the collision.

To solve this problem, we can apply the principle of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision.

The formula for momentum is given by:

Momentum (p) = mass (m) * velocity (v)

Before the collision, Alan's momentum (p₁) is equal to 60 kg * 4.0 m/s = 240 kg·m/s.

Before the collision, Betsy's momentum (p₂) is equal to 50 kg * (-0.4 m/s) = -20 kg·m/s (since her velocity is in the opposite direction).

After the collision, Alan's momentum (p₃) is equal to -60 kg * 2.0 m/s = -120 kg·m/s (since his velocity is in the opposite direction).

Let's assume Betsy's velocity after the collision is vf.

To find Betsy's velocity (vf) after the collision, we set up the equation for conservation of momentum:

p₁ + p₂ = p₃ + m₂ * vf

Plugging in the values:

240 kg·m/s + (-20 kg·m/s) = -120 kg·m/s + 50 kg * vf

220 kg·m/s = -120 kg·m/s + 50 kg * vf

Rearranging the equation to solve for vf:

50 kg * vf = 220 kg·m/s + 120 kg·m/s

50 kg * vf = 340 kg·m/s

vf = 340 kg·m/s / 50 kg

vf = 6.8 m/s (rounded to one decimal place)

Therefore, Betsy's velocity after the collision is 6.8 m/s.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.

First, let's calculate the initial momentum of Alan and Betsy before the collision:

Initial momentum of Alan (p₁) = mass of Alan (m₁) × velocity of Alan (v₁)
= 60 kg × 4.0 m/s
= 240 kg·m/s

Initial momentum of Betsy (p₂) = mass of Betsy (m₂) × velocity of Betsy (v₂)
= 50 kg × 0.4 m/s
= 20 kg·m/s

The total initial momentum (p_total) before the collision is given by the sum of the individual momenta:

p_total = p₁ + p₂
= 240 kg·m/s + 20 kg·m/s
= 260 kg·m/s

Now, let's consider the final momentum after the collision. After the collision, Alan is moving backward at 2.0 m/s (taking positive direction as his initial velocity). Therefore, his final velocity (vf) is -2.0 m/s.

Using the principle of conservation of momentum, the final momentum (p_final) is equal to the total initial momentum:

p_final = p_total
= 260 kg·m/s

Final momentum of Alan (p₁f) = mass of Alan (m₁) × final velocity of Alan (vf)
= 60 kg × (-2.0 m/s)
= -120 kg·m/s

Final momentum of Betsy (p₂f) = mass of Betsy (m₂) × final velocity of Betsy (vf)
= 50 kg × vf

Since the total final momentum is equal to the initial momentum, we can write the equation as:

p₁f + p₂f = p_final

-120 kg·m/s + 50 kg × vf = 260 kg·m/s

Rearranging the equation:

50 kg × vf = 260 kg·m/s + 120 kg·m/s

50 kg × vf = 380 kg·m/s

Dividing both sides by 50 kg:

vf = 380 kg·m/s / 50 kg

vf ≈ 7.6 m/s

Therefore, Betsy's velocity (vf) after the collision is approximately 7.6 m/s.