Here is the full list of all 21 Project Root (JP) trophies - there are 21 bronze.
Level 2 complete
Kill 50 Tanks
Destroy 500 missiles
Travel 100 kilometers
Level 1 complete
Kill 100 Enemies
Tutorial complete
Level 3 complete
Level 4 complete
Level 5 complete
Level 6 Complete
Level 7 Complete
Easy game complete
Normal game complete
Hard game complete
Die 50 times
Collect 10 lives
Collect 50 Homing Missiles
Play 5 hours
Complete level 1 in less than 4 minutures
Complete level 5 without losing a life