Purdue will be close but slightly above your budget (mid 40s). But they have a competitive racing team:
However, it’s also in a rural location.
Another school you might want to consider is UMD. Like Purdue, they too have a strong mech engineering program as well as astronomy. But they’re in a suburban area just outside of DC, and they also have racing teams:
Although their COA is much higher, your son may get merit based on his profile that could get the cost down to the mid 40s (if you can stretch a bit).