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Do the boots of speed affect my jump distance? I'm not sure because it refers to increasing my walking speed, and not movement speed.

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Essentially No

"Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of movement." Page 182 of the PHB

So basically I interpret that as no matter how much movement you have you can only jump as far in feet as your strength score.

There are some exceptions though:

Second Story Work

When you choose this archetype at 3rd level, you gain the ability to climb faster than normal; climbing no longer costs you extra movement. In addition, when you make a running jump, the distance you cover increases by a number of feet equal to your Dexterity modifier. Page 97 PHB

So at third level rogues who take the thief archetype can add their dex and their strength to the jump.

Step of the Wind

You can spend 1 ki point to take the Disengage or Dash action as a bonus action on your turn, and your jump distance is doubled for the turn. Page 78 of the PHB

so monks can make their jump distance strength * 2 for one turn by spending a ki point at second level.

Remarkable Athlete

Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Dexterity, or Constitution check you make that doesn’t already use your proficiency bonus. In addition, when you make a running long jump, the distance you can cover increases by a number of feet equal to your Strength modifier. Page 72 of the PHB

So fighters who take the champion archetype can add their strength to the jump distance making it strength score + strength modifier at seventh level.

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No.

First, your "walking speed" is your speed. See PHB p181, the first sentence under Speed:

Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.

So it doesn't matter that it's not referring to "movement speed". The game term in 5e is "speed".

However, your jumping distance is not affected by your speed, per PHB 182:

Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of movement.

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  • \$\begingroup\$ Re: your "however" line, there are those who argue that your jump distance is affected by speed, in that it caps the jump. See this question. \$\endgroup\$ – nitsua60 Nov 29 '17 at 1:04
  • \$\begingroup\$ How do you account for different movement modes? Creatures can have a walking speed, a flying speed, a swimming speed, a climbing speed, or a fly (hover) speed. Boots of speed explicitly only affect "walking speed". \$\endgroup\$ – keithcurtis Nov 29 '17 at 1:49
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    \$\begingroup\$ @keithcurtis Look at monsters. The walking speed is always defined as the untyped movement under Speed. Look at the race descriptions. Walking speed is always defined under an ability called "Speed." Look at Aarakocra. Their flying speed is not listed under speed. Look at a character sheet. It says "speed" not "walking speed." "Speed" by itself is essentially synonymous with "walking speed" in game terms. Other movement modes always explicitly call themselves out. That's the point I'm making, although perhaps I'm coming at it from the wrong direction. \$\endgroup\$ – Bacon Bits Nov 29 '17 at 13:17
  • \$\begingroup\$ I understand your point, but the answer gives the impression that one movement mode is unaffected by another. You can't for instance move your full walking speed, dash your full dash and then jump 10 feet. Your jump is counted out of speed for the turn. (Per the section on jumping). IOW, Jumping, crawling and swimming (assume no Swim speed) come out of your walking "budget", but since the boots go out of their way to specify "walking speed" instead of just "speed", they should not be assumed to enhance other modes. I just felt that the answer as written was unclear or misleading difference. \$\endgroup\$ – keithcurtis Nov 29 '17 at 20:36
  • \$\begingroup\$ @keithcurtis I think you're trying really hard to read too much into it. \$\endgroup\$ – Bacon Bits Nov 29 '17 at 20:42

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