Is expiation obligatory for the aforementioned oath, and would it be one expiation or two, and what are the methods of expiation?

If you did not ride the car, nothing is incumbent upon you. However, if you did ride it, you have broken your first oath, and expiation becomes incumbent upon you. Consequently, the second oath is also broken due to the obligation of rendering the expiation.

Therefore, two expiations are incumbent upon you. The expiation for an oath is feeding ten poor persons (approximately 750 grams of food for each poor person), or clothing them, or freeing a slave. If you are unable to do any of these, you should fast for three consecutive days, as a precautionary measure.

Allah the Almighty has expounded on this in His saying: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or freeing a slave. But whoever finds not [the means] - then a fast of three days. That is the expiation for your oaths when you have sworn. And guard your oaths. Thus does Allah make clear to you His verses that you may be grateful." (Al-Ma'idah: 89).