Push is a wretched spell. And not simply because it's totally "weak sauce" as an offensive choice for the starting, first level magic-user (compared to, say, sleep or the dual-purpose light spell). No, it is wretched because of the way it's written, in game terms, especially when compared to other spells of similar effect.
[oh, hi! Remember me? Yes, still alive here at in plague-ridden Seattle. Currently a couple dudes of Eastern European persuasion are working on fixing my dryer while the family sleeps away upstairs]
Yesterday, I spent far too long on researching joules and newtons and physics calculations to figure out the correspondent scale between the push spell and the 5th level magic-user spell telekinesis. Just so I don't have to go through all that again, I'm going to write it up here for edification of interested parties. Because the phrase
"Heavy objects travelling [sic] at high speed can be deadly weapons!"
- PHB, page 82
isn't especially helpful in and of itself. How heavy? What speed? How many hit points of damage? And how does it compare to push, which simply exerts one foot-pound of energy, per level of the caster?
Let's start with telekinesis: the spell moves 25 pounds of weight, per level of the magic-user. As would be expected from a fifth level spell, control is much more precise than for the push spell, and while the caster must concentrate to control the object being moved, the duration only lasts for two rounds plus one additional round per level. Speed starts at 2" (20 feet) per round, and then doubles with each successive round until a maximum of 1024" (10240 feet) in the tenth round. As magic-users first gain the ability to cast a fifth level spell (like telekinesis) at 9th level, I can see that at minimum the caster will be able to accelerate 225 pounds to the maximum speed by the tenth minute of concentration, and maintain control at that speed for one additional minute (each round being one minute in length).
Joules are the measure of kinetic energy and is a unit found by multiplying half an object's mass by its velocity squared in terms of meters per second (m/s). Assuming gravity in our D&D setting is the same as real world Earth, 225 pounds is equivalent to 102.058kg. Converting D&D "inches" (tens of feet) per round to miles per hour...and thence to m/s...we can calculate the telekinetic velocity over time as follows:
1st round: 0.1016 m/s
2nd round: 0.2032 m/s
3rd round: 0.4064 m/s
4th round: 0.8128 m/s
5th round: 1.6256 m/s
6th round: 3.2512 m/s
7th round: 6.5024 m/s
8th round: 13.0048 m/s
9th round: 26.0096 m/s
10th+ round: 52.0192 m/s
[for my American readers, that's a bit more than 116 miles per hour at maximum velocity]
Here's a good web site for calculating kinetic energy (in joules). Suffice is to say that a 9th level magic-user using telekinesis uses about half a joule in the first minute and quickly ramps up, generating a bit more than 8 joules after three rounds, 33 joules after four, and nearly 135 by the round five. At maximum velocity, that 225 pounds is using over 138 thousand joules of kinetic energy.
Meanwhile, the same 9th level magic-user using push generates only nine foot-pounds of kinetic energy (one foot-pound per level): a little more than 12 joules of kinetic energy.
Or does it? Let's take a closer look.
"Of course, the mass of [the target] cannot exceed the force of the push by more than a factor of 50, i.e. a 1st level magic-user cannot effectively push a creature weighing more than 50 pounds."
- PHB, page 68
Okay, just because it drives me crazy, I'm going to go ahead and convert Gygax's pounds to kilograms (because you measure mass in kg, not pounds). If we're going to say that a 1st level magic-user can move 22.6796kg a single foot by means of the push spell, and that this is a single foot-pound of kinetic energy, then working backwards we can discover that the "instantaneous" duration found in spells like push can be measured in actual time as .8814 seconds.
[the PHB states "instantaneous" means a spell "lasts only a brief moment." Strange fact: did you know that a moment was once an actual unit of time, roughly calculated to be 90 seconds? The things you discover...]
Now we look at the 9th level magic-user using push to exert nine foot-pounds of kinetic energy (12.2024 joules) with the spell. Knowing the time, distance, and KE we can determine the mass that can be moved a single foot in .8814 seconds as 204.092kg. Converting that to the maximum amount of weight that can be pushed we see it's (roughly) 449.95 pounds. Pretty close to 50 pounds per level (which is what the text implies).
But what if my 9th level sorcerer wanted to push 175 pounds instead of 450...say, the weight of an average human male (DMG page 102)? How about 60 pounds, average weight of a male halfling? Well we can see the velocity created by the kinetic energy will change in these cases: specifically to 0.55448 m/s (for the human) and 0.946955 m/s (for the halfling). Knowing this, we can calculate the human will be pushed with enough force to travel more than two feet, while the halfling will be pushed three and a half. Judging by how far I can knock my 60 pound child in play, I exert more foot-pounds of force than this.
Now, if the sorcerer launched a dart (.5 pounds; equivalent of .2268 kg) from a flat surface (say, the palm of her hand) it would travel about 39 feet...nearly the same distance as the weapon's long range. Would a magically impelled dart lose velocity over distance like a thrown dart? Yeah, probably (since the initial impetus of force is at the place the dart is initially resting). But, still, that's kind of a neat trick.
Not as neat as magic missile, of course, which shoots five unerring darts at 9th level. Probably not even as effective as simply throwing three darts per round (with no chance of being "interrupted"). Yeah, I guess it's not really neat at all.
Push is just a wretched spell. It's the equivalent of a cantrip. I want my first level spells (my first level offensive spells certainly!) to be effective attacks. As written, it should be a defensive spell seeing as how a target ends up with a penalty to its attack roll. If it fails its save. If.
Personally, I'd prefer something that really shoves something...smashes targets against walls or flings them over cliffs. Right now, this spell is just a nudge. And that's not good enough.
I'll talk about the detritus that is gust of wind another day.