Speed up means to start going faster, either by walking faster or driving your vehicle faster. You can also speed a process up by taking actions to make it go faster, or speed a person up, by pressuring them to move faster.
Ex: As we were driving down the street, several cars pulled up close behind us and honked their horns. They clearly wanted to hurry us a bit, so we decided to speed up, even if it meant speeding a little.
Ex: As the students were writing their essay exams, their teacher looked at the clock and saw that time was almost up, so he told them to speed up their writing a little.
The opposite of speed up is slow down. When you want someone to finish a task more quickly, you can just say speed it up as a synonym of the expression "hurry up!"
Ex: When I looked at the speedometer, I realised I had been speeding, so I took my foot off the accelerator a bit to slow down. I didn't want the police to pull me over.
Ex: Lisa always takes a long time to get ready. She takes a long shower, thinks about what outfit to wear, spends an hour putting on her make-up. She really needs to speed it up if she wants to start working in the mornings.
To accelerate is to increase in speed, usually in an official way. You can say speed up in most everyday situations, but accelerate in more official or scientific contexts. You can also accelerate an event, which is making it happen faster.
Ex: The law says that accelerating beyond the speed limit even if only for a few minutes is not allowed. It will be considered speeding.
Ex: The banking crisis accelerated the process of inflation in emerging market economies like Russia. It was like the late 1990s all over again, only it happened faster this time around.
A school program that is faster or more intense for smarter students is often called an accelerated program. Another word for gas pedal, or where you put your foot to make a car go faster, is the accelerator.
Ex: My niece is super smart for her age. Her teachers put her in an accelerated program for math and science last week, so she'll be doing math with kids much older than her by this time next year.
Ex: Ciaran realized he was going pretty slow, so he decided to speed it up and stepped on the accelerator. He was surprised by how quickly the car started to move.