1. We mentioned yesterday that one should not run on Shabbat even when not exercising since one’s walking on Shabbat should be different than the rest of the week. However, running for mitzva purposes is permitted. Accordingly, one who wishes or needs may run to shul (Shulchan Aruch, O.C. 301:1). One may also certainly run if one needs to reach a child or family member who needs immediate assistance.

2. Young children who enjoy jumping and running are permitted to do so (Shulchan Aruch, O.C. 301:1). This includes playing games involving running and jumping, though actual exercise on Shabbat is not necessarily recommended even for them (Shemirat Shabbat K'hilchatah 16:40, though see n.106 that it may be permitted if they enjoy it).

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Dedicated by Fran Broder as a zechus for everlasting peace in Eretz Yisrael in the merit of learning Hilchos Shabbos.