# Snow shovelling physics

Writing the how-to shovel article recently got me thinking can we actually quantify that the method is more efficient?

After thinking about it a little bit, with some assumptions I think yes!

Disclaimer – I just had a baby (3rd) and recovering from surgery so heavily sleep deprived, there may be mistakes here! Please point them out in the comments.

TLDR version -> pushing snow takes less effort/energy than lifting and throwing it. So to be most efficient from an energy consumption point of view (and to make it easier) push the snow as much as you can before lifting/throwing! The further you are when you throw, the more energy and power it takes.

A few words to clarify things:

• Energy – the capacity to do work (ie you use energy to do work)
• Work – Force multiplied by distance
• Force – the amount of strength you exhert to do something (effort)
• Power – how quickly you do work
• Efficiency – useful work divided by total energy expended

Often I get comments that my shovelling technique (see article here) is too hard or takes too much energy. I think people don’t understand why this technique works and what makes it actually easier and take less energy. People often get work and power mixed up.

Think about it this way, if you decide to do a marathon, you can walk it or you can run it. If you run, you will need a great deal more power since you will finish much more quickly. But in terms of useful work (travelling the marathon distance) there is no difference between walking and running, you will do the same amount of useful work.

The ultimate goal for everyone, is to shovel the snow from their driveway onto their lawn. There are only really 3 ways you can do this:

• Push the snow until you get to the lawn
• Pick up the snow and throw it onto the lawn
• Pick up the snow and walk with to your lawn and dump it

### Pushing the snow

Pushing the snow requires the least effort, you are only moving the snow in one direction. Basically just countering friction between the snow/shovel and the ground. If there is a lot of snow, perhaps you are compacting it a little bit which does take some more effort. The equation will look something like this:

Force = friction coefficient (<1) * Normal force (weight of snow being pushed)

Energy will be simply Force*distance pushed

The force also increases as you push the snow, more snow is being pushed so the heavier it gets. Your push is easy when you start, and harder as you gather up snow. If you move slowly, you don’t need a lot of power but if you decide to do it quickly you will need more power and get winded faster.

However, if there is too much snow, you might not have the strength to keep pushing it, at that point you don’t really have a choice but to lift it up and throw it or walk it to the snowbank.

### Pick up snow and throw it onto the lawn

This requires a bit more effort (and more power). You’re not only moving snow towards your lawn, you must also lift it up (countering gravity) and by throwing it, you need to make sure it won’t hit the ground before it goes where you want it! We can use the following equation to calculate energy:

Energy will be 0.5 *mass * velocity * velocity (or 0.5*mass * velocity squared)

Keep in mind here that the further you are from the snowbank, the more energy you need to counter gravity and get it there. To get snow onto the snowbank (assuming it is the same height as your shovel) the total distance in the vertical direction must equal 0 and the final vertical velocity must equal to the initial vertical velocity (but in the opposite direction). So essentially acceleration of gravity * time must equal to 2 * the initial vertical velocity (1 to get down to 0, 1 to get it equal to the initial vertical velocity in the other direction)

0 = 2*Velocity of snow (vertical) – acceleration of gravity * time of flight

isolating for time of flight (how long the snow is in the air):

Time of flight = 2*Velocity of snow (vertical)/acceleration of gravity

The velocity in the horizontal direction must also span the distance between you and the snowbank within that time of flight. So we have

Distance to snowbank = Velocity of snow (horizontal) * time of flight

The optimal angle to throw the snow would be 45 degrees -> Sin(45) for the vertical direction and cos(45) for the horizontal direction are maximized this way, both are equal to 0.707 of the total velocity.

If we combine those equations where

Velocity of snow (horizontal) = Velocity of snow * cos(45) and

Velocity of snow (vertical) = Velocity of snow * sin(45) we have:

Distance to snowbank = Velocity of snow * cos (45) * 2 * Velocity of snow * sin(45) / acceleration of gravity

which we can combine/simplify to:

Distance to snowbank = Velocity of snow * Velocity of snow / acceleration of gravity

and plugging in some numbers – if you are 1m away, Velocity of snow = 3.13m/s

but if you are 5m away, Velocity of snow = 7m/s

If we look at the energy equation – Energy = mass * 1/2 velocity * velocity, so essentially throwing the same snow from 5 meters away takes 5 times more energy than throwing it when you are 1 meter away

In other words, get as close to the snowbank as you can to reduce the amount of energy you need to get it there.

### Pick up the snow and walk with to your lawn and dump it

Now this one is a bit more complicated, with the previous section we can assume a lot of things but energy consumption while walking and its relation to weight is not so simple to calculate (nor is it simple if you are walking in deeper snow). Overall, I think we can assume it likely takes less energy than throwing it in most cases but it will take quite a bit more time. I usually don’t do this unless the snow is super heavy since it is just too time consuming!

So essentially push the snow as much as you can, throw it (or carry it) onto the snowbank is the best way to do it!

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