The poll for road safety charity Brake among 1,665 children found 13 per cent of 11- to 13-year-olds and 6 per cent of nine- to 11-year-olds had been involved in such incidents.
In addition, more than half of those questioned said they had been involved in a near miss.
A similar proportion felt the roads around their home and school were dangerous for ped-estrians and cyclists, with 87 per cent believing vehicles travelled too fast in their area.
Brake, which published the poll to coincide with road safety week, described its findings as "outrageous".
Fraser Simpson, a campaign spokesman for the group, said: "All drivers have a part to play in making our roads safer for children, and one of the best ways we can do this is to slow down to 20mph or below around schools and homes.
"If you do, you have a good chance of being able to stop in time if a child runs out in front of you, and you're helping to make our communities greener, more pleasant, and more family-friendly.
"Children in Scotland understand the need for traffic to slow down, and are appealing to drivers to make this simple commitment."
Five children were killed on Scotland's roads last year, compared to 20 in 2008. Deaths this year include a five-year-old boy being hit by a lorry in Gullane, East Lothian, in August.
Transport Scotland said there were 20mph limits at most schools. A spokeswoman said: "Between 2003 and 2008, the Scottish Government provided grant funding of nearly 50 million to local authorities to implement 20mph speed limits around schools and related safety projects.
"Additional guidance was also provided in 2004 to assist local authorities in implementing 20mph speed limits around schools on roads with speed limits higher than 30mph.
"By the end of March 2008, when the funding stream ended, local authorities reported 20mph schemes were in place around 83 per cent of schools."
• Primary school children of any age cannot accurately judge vehicle speeds above 20mph, researchers at Royal Holloway, University of London, report today. Professor John Wann, who led the study, said it emphasised the need for drivers to slow down.