1. Analyze the Physics of Snow Accumulation

Snow is falling vertically. When it lands on the sledge, it has zero horizontal velocity. The sledge must exert a force to accelerate this snow to the sledge’s current velocity $v$. By Newton’s 3rd Law, the snow exerts a drag force backwards on the sledge.

This “drag” slows the sledge down. The strategies involve how we manage this mass.

2. Analyze Strategy II: Sweep perpendicular to track (frame of track)

Action: You sweep the snow such that it leaves the sledge perpendicular to the track as seen by someone on the ground.

Physics: This means the snow leaves with horizontal velocity $v_x = 0$. Essentially, you stop the snow’s forward motion immediately after it lands. The sledge pushes the snow backward to stop it; the snow pushes the sledge forward (thrust).

Ideally, if you deflect the snow perfectly such that it never gains forward momentum (relative to the ground), the sledge loses no momentum to the snow. The mass of the sledge remains constant (assuming steady sweeping), and since no momentum is lost to the snow, the velocity remains constant (or decreases the least). This is the Best Strategy.

3. Analyze Strategy III: Do Nothing

Action: Let the snow accumulate.

Physics: Conservation of Momentum: $P = (M_{sledge} + m_{snow}) v$. Since $P$ is constant (no external horizontal forces) and mass increases linearly with time ($M(t) = M_0 + rt$), velocity must decrease.

$$v(t) = \frac{P_{initial}}{M_0 + rt}$$

The velocity decreases proportional to $1/t$. This is the Intermediate Strategy.

4. Analyze Strategy I: Sweep perpendicular to sledge (frame of sledge)

Action: You push the snow sideways relative to you.

Physics: Even though you push it sideways, the snow still retains the forward velocity $v$ of the sledge (since you didn’t push it backward relative to the ground). So, the snow leaves carrying away momentum equal to $dm \cdot v$.

Here, you pay the “drag penalty” when the snow lands (slowing you down), but you don’t get any “thrust bonus” when throwing it off because you don’t throw it backwards. You just reduce your mass.

The equation of motion leads to exponential decay of velocity: $v(t) = v_0 e^{-\frac{rt}{M}}$.

Since exponential decay drops faster than the polynomial decay ($1/t$) in Strategy III, this results in the lowest speed. This is the Worst Strategy.

5. Conclusion

Ordering from Best to Worst:

1. Strategy II (Best – maintains momentum best)
2. Strategy III (Middle – velocity decays as $1/t$)
3. Strategy I (Worst – velocity decays exponentially)

Order: II, III, I.