- Open your picture
- type the text in the font, color and size you want and make a smart object of it (to be able to change applied filters)
- place the text on his place by using warp
- apply path blur in the blur gallery
You can change the speed and taper as you wish.
As it is a smart object you can go back to the applied filter to make changes.
Here's a tutorial on path blur:
https://www.youtube.com/watch?v=vcTMQTzVpDI
Another way to do this is by using motion blur.
But since you have to set your blur direction make the text first without changing it, turn it into a smart object. Apply the motion blur 0° on the text as it is.
Once applied move the text to his place and warp it on his correct place. Warping it into a curve will assure that the blur direction stays in the curve of the text.