Can a police officer search my car when I am stopped for speeding?

Generally, no. However, due to the mobility of cars, time does not permit a search warrant to be obtained. As a result, vigilant police are motivated to search suspicious automobiles. However, the police officer must have probable cause to believe that contraband is concealed somewhere. 

In essence, merely being stopped for speeding should not allow the officer to search your car; however, if the officer saw you throw an empty beer can out the window, that may be sufficient probable cause to search your car. Or, if the officer smells marijuana as he approaches the car, he may have an articulable suspicion to search.

It is unreasonable to make a search of an automobile when the arrest is for a minor traffic violation (like speeding), as a subterfuge for a search for evidence of a serious crime. Yet, the many automobile exceptions are based on the lower expectation to the right of privacy in a car versus a home and the fact that cars are mobile and evidence can be more readily disposed.

In addition, while it might seem unfair, an officer is allowed to search a car based on a pretext. This means that if you look suspicious to the officer, he can find any reason to pull you over, then look for probable cause while he talks to you on the side of the road. The Supreme Court has stated that it doesn't matter how flimsy the initial stop was, if the officer then has probable cause, the resulting search is legal.

There is always the "officer safety" exception as well. If the officer says he saw you bend over as if to hide something under the seat and that he feared it was a gun, then he would be upheld by most judges as reasonably protecting his own safety during the speeding stop. If so, any drugs he found instead would be legally used against you.