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Is there anything a character can do when shot at with a missile/grenade? If said grenade wasn't set to explode immediately, they can try to catch it to toss it back (REF+COO+COO), or jump on it (REF+COO+WIL). It can perhaps even be argued that they can use Fray as usual (divided by half) when literally dodging a missile that flies past them.

Can Fray be used to evade the damage somehow, like jump for cover or just further from the explosion? From the description, it seems like this is not an opposed test, and instead the attacker has to contend with scatter, though the rules are vague as ever.

To give a concrete example, say an enemy has just tossed a grenade at my character, rolling 45 - under their Throwing Weapons skill of 50. What happens next, by the rules?

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I'd say quarter of Fray for dodging an unguided missile or grenade, as half is for guided projectiles or bullets/lasers which would be much harder to dodge. I'm not a fan of the multiple aptitude rolls in the core book and just simplify them to aptitude x3.

Explosions decrease in damage by 1 DV every meter you're away from the center of the explosion, so it wouldn't be opposed. I'd use a Fray/Freerunning/Flight/Freefall/Swimming roll , with MOS determining how far away they got before the explosion.

With your throwing example it would be that they throw it right at your character, getting within full DV range. An Opposing Fray roll would be to dive away to minimize damage or get behind cover. So, if the defender succeeded but got lower then they'd dodge away but still take damage, if they won they'd have gotten to cover(Just make some up) and avoided the effects of the blast.

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Move it

The key to reducing damage from the explosion is moving away from it or moving behind cover, and how doable that is will depend on the interaction of movement rules with the time and position of the explosion.

Instant Explosions

Eclipse Phase's native system (unless I'm missing a rule outside the core combat ruleset) doesn't have a rule that allows reflexively taking a movement quick action on an enemy's turn as you dodge. (This contrasts with systems such as GURPS, where a character may usually move one yard when dodging an enemy attack.)

As a result, there's very little that can be done if a grenade or seeker is instant-detonation (airburst, impact, or even signal triggered mid-flight). This is actually quite plausible: even thrown grenades tend to move rather fast, and the time between it passing a (missed) direct hit point and reaching the detonation distance, getting a detonation signal, or impacting the nearby surface, is likely to be in the hundredths or at most a couple tenths of a second; meanwhile, a even a transhuman morph just can't even accelerate to a necessary speed to move an appreciable distance away in that time frame.

So be glad if you at least managed to avoid a direct impact, and make sure you're not late on your morph insurance payments.

Delayed Explosions

Things get drastically better for the defender in this scenario.

Eclipse Phase corebook, page 200

The minimum detonation period—1 second—means that the munition will detonate on the user’s (current) Initiative Score in the next Action Phase.

In this case, the ability or inability to reduce or avoid an explosion emerges naturally out of the movement, speed and initiative rules.

If, after the grenade was thrown, the defender is eligible to perform more actions, it can use the next such opportunity to make a run for it.

  • If the defender has a lower initiative than the attacker, then it can move out of the way during the same Action Phase.
  • If the defender's initiative is better than the attacker's, but the defender has enough Speed to act the next Action Phase of the same Action Turn, then it can get out of the way during the aforementioned next Action Phase in this Action Turn.
  • If the grenade was thrown in the last Action Phase of the Action Turn, then the next Action Phase occurs in the next Action Turn, and the ability to take a defensive action before the grenade explodes will depend on the initiative roll in the next Action Turn.
  • If the defender lacks the Speed to act in the next Action Phase of the Action Turn, then the grenade falls to the ground (wherever the scatter indicates), and explodes too soon for the defender to get out of the way.

Just how far the defender moves out of the way will of course depend on the movement method chosen (walk, run, jump), obstacles in the way etc.

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