From the excellent ' The Insect That Stole Butter' the excellent Oxford Dictionary of Word Origins.
Under 'Raj'

From the same dictionary
“crop [OE] From around AD 700 to the late 18th century crop, related to group [L17th], had a sense 'flower head, ear of corn', which gave rise to the main modern meaning 'a cultivated plant grown on a large scale' and also to senses referring to the top of something, such as the verb uses 'to cut very short' or 'to bite off and eat the tops of plants'. The sense 'a very short hairstyle' goes back to the late 18th century but is particularly associated with the 1920s, when the Eton crop, reminiscent of the style then worn at the English public school Eton, was fashionable for young women.
To come a cropper is to suffer a defeat or disaster. The origin of the phrase maybe the 19th-century hunting slang term 'cropper', meaning 'a heavy fall'. Cropper probably came from neck and crop, an expression meaning 'completely or thoroughly' and originally used in the context of a horse falling to the ground. Crop here referred either to the rider's whip (originally the top part of a whip) or the horse's hindquarters. This sense is found in Old French croupe 'rump', which appears as croup in Middle English, and is the source of the crupper [ME], the bit of harness that goes from the saddle under the horse's tail, and which lies behind the word croupier [E18th]. In early use, this was a term for a person standing behind a gambler to give advice, adopted from French, cropier 'pillion rider, rider on the croup'.â€쳌

Eric Partridge’s Dictionary of Historical Slang confirms this:
“Cropper; esp. come, or go a cropper. A heavy fall, fig. or lit.: from the late 1850s; coll.
Trollop, 1880, “he could not… ask what might happen if he were to come a cropper.â€쳌
Ex. Hunting.â€쳌