Phrase(s): race for something
to run or drive fast to get to something in a hurry. • I raced for the door to see who was there. • The children raced for the best seats in front of the television set., Phrase(s): race someone for something
to compete against someone for a prize; to try to outrun someone to get to something first. • I will race you for the grand prize. • Ned raced his sister for the breakfast table every morning.
McGrawhill's American Idioms And Phrasal Verbs