How much should a 10-year-old trumpet player practice daily?
As in all such things, it depends. Such guidance should really be given with the advice of the kid's music teacher, who knows a lot better than I what kinds of material are being practiced, what skill level the kid has reached at this point, and so on.
For instance, 10 years old put a kid roughly in the area of 5th grade, which likely means they very likely have no more than one year of trumpet playing under their belt. That's not a lot. If they started in 4th grade and have been playing a year, then twenty minutes a day is probably reasonable. If they're just getting started this year, then twenty minutes a day is probably more of a goal to reach by the end of the year than a perfect starting point; for beginners, fifteen minutes is probably a good starting point. I would try to get to twenty minutes by the end of the year, and then try to reach 30 minutes a day by the time they've reached junior high. If they're still playing in high school, and if the kid is showing real interest in playing, then the target should be an hour a day. And all this might well be tossed out if it turns out that the kid has some serious passion for music, because then they'll want to practice more than any arbitrary goal anyway.
Studying a musical instrument is something that requires discipline, but it also requires motivation. It's easy for practice, especially for beginners, to settle into a drudgery that's just something to get through as quickly as possible. For most kids, music lessons start as a social thing, in school, as part of a band; the act of sitting in a room by oneself "practicing" isn't really something that a 10 year old kid has a great deal of experience with, so it's hard for them to get the habits down, and learn how to practice. My own experience was that I spent the better part of a year or two just flailing around, not really practicing much, until teachers and my older sister took some time and showed me how to practice. Giving a kid a piece of music and saying "Work on this" is well and good, but if the kid doesn't know how to work on it, then practice time becomes a boring exercise in playing through a piece badly a few times and then putting the horn back in the case.
I had a teacher once who had me working on some concerto or other -- maybe the Haydn, maybe the Arutunian -- and he said to me, "You can play this concerto just as well as Maurice Andre can. (Maurice Andre was one of the greatest trumpet players of all time.) You can play every note just as cleanly and precisely as he can. You know what the only difference is? He can do it faster than you can." This teacher taught me to slow a piece down to the point where I could play it perfectly, even if that meant slowing it down so much that it was unrecognizable. Then, he said, gradually increase the speed. If you find you can't play it perfectly at a certain speed, back it off again and work at it until you can. Eventually you get it so you can play it up to tempo. (And at the same time, this process develops all the various skills along the way so the next piece won't take so long to get to tempo.) That's what I mean by teaching how to practice. This guy, Mr. Rudgers at the Bristol Hills Music Camp, showed me a good way to work at a piece. It helped.
Another thing is that you never know if, or when, the motivation is suddenly going to strike. A kid might frankly suck at the trumpet for two, maybe even three, whole years before they wake up one morning and decide that they'd rather not suck anymore, or that Wow, that other kid is really good and hell, I can be just as good as that kid if I work at this and hey, maybe practicing isn't so bad in the first place. I know that can happen, because that's the way it was with me: I was a shitty trumpet player and got made fun of relentlessly by the other kids in band because I was shitty until I decided "OK, I am now going to work on not being shitty." That's not a decision you can make for your kid. Eventually everybody decides they're going to be good at something. If it's music, great. If not, then at least they'll hopefully learn enough to have a greater appreciation for music in the future. And besides, a person can be passionate about more than one thing. My active life in music is long over, and occasionally that's a source of rare regret for me, but had I stayed in music, it well may be that a couple of princesses never take to the stars. Who knows.
I'd also suggest that parents should not treat practice time as a daily home recital. All the parent should really do is enforce the agreed-upon duration of the practice session, and that's about it. Unless you're a musician yourself, don't say anything, other than an occasional "Hey, I can hear that your sound is getting a lot better!" Do not say anything like "Gee, Timmy, that one piece you played sounds like you need to work a lot more on it." Believe me, Timmy knows, and shame is not the emotion you want Timmy carrying into his interactions with music. And even if you do know a lot about music...back off anyway. It's really for the best. Parental expectation is also not a great thing to have to struggle with when you're also trying to remember the fingerings for D-natural and A-flat. (Along these lines, unless the instrument is the piano and therefore the thing is wherever it is, let the kid practice in their room or some other room with a door. Don't make them practice in the living room while you're there paying bills or watching the evening news or making dinner in the adjoining kitchen. Practicing should not be done with an audience, and the only reason practicing should be heard by parents at all is to know that it's getting done.)
Let's see, what else? Oh, yeah -- when I say "twenty minutes a day" or "an hour a day", I don't mean each and every day. Practicing seven days a week isn't wise, in my opinion. Now, passion may arise and then the kid will practice every day because they love playing, but even then, I'm of the view that a day off is good. It's good mentally, for what I hope are obvious reasons, but it's also good physically. Playing an instrument involves the use of muscles, and in quite a few cases (especially among the wind instruments), the muscle use is strenuous indeed. If you don't believe me, watch a great trumpet player sometime, or a horn player, or an oboe player, or any instrument. Making air vibrate the way it's supposed to inside a wind instrument requires making the muscles of one's face and neck do things that they don't normally do. Those muscles are collectively referred to by the word embouchure, and like all such muscle groups, they can be overstressed, injured through misuse, and worn out. The effects of a hard practice session or rehearsal on a trumpet player's embouchure are similar to the effects on a weight-lifter's muscles after a lifting session, so for the exact same reason, wind players should take a day off here and there, or if they do play, it should be something low-stress, like long tones in the low register. That's more the equivalent of stretching than exerting. I always found, though, that after a week of playing several hours a day (between group rehearsals and practice sessions), taking a day to not play at all (often Saturdays while I was in college) made me a lot stronger when I played again on Sunday.
For a 10-year-old, it's probably best to set a daily time as practice time, as much as that's possible. Eventually it will hopefully become sufficiently ingrained that they'll practice on their own, but to start with, the expectation that every day at, say, 5:30 they are to go practice for their fifteen or twenty minutes will help. Maybe let them not practice on Fridays or Saturdays, or some other weekday if they're on a soccer team or something like that.
And finally, maybe try not calling it practice. I once knew a drummer who said, "I don't practice. I play." The word "practice" just has the sound of required daily boring routine. Saying "Hey, Timmy, it's 5:30! You need to go play your trumpet for a while!" simply sounds better than "Hey, Timmy, it's 5:30, you need to practice." Practicing sounds like what you do until you're good enough to play. Maybe we'd have better luck with our kids and their music lessons if we enforced the idea that it's all an act of play. I have a notion that we call it practice because in that way we can trick our ever-so-Puritan American minds that we're not actually wasting time learning to play music, and I say, the hell with that. It's all play. Playing is good.
So let your kid play.