Optional objective: Neutralize platoons with a single cannonball - 8.
Optional objective: Kill 2 platoons using a single cannonball.
Optional objective: Completed all the constraints in one playthrough.
The beginning of the missions focuses on shooting the cannon into the approaching British soldiers. Use the left analog stick to aim and the jump button to shoot. Always be sure to eliminate the enemies marked with red dots in the first place. They are the closest and might hurt your units. To end the mission, you can't lose all of your soldiers before the time ends.
To get a full synchronization, try to make the aim circle embrace a whole platoon. Fire eight times like that and the first objective will be completed. It will be a bit harder to destroy two platoons with one cannonball. You should try to embrace two platoons standing in one row for that. Also remember that the two platoons don't have to be complete.
Optional objective: Prevent patriot executions - 3.
As you're done with the cannon, run forward and jump onto the nearby horse. You will have to ride it from one green point to the other until the mission ends.
For the optional objective, you will have to keep an eye on the surroundings. You should be passing by three groups of British soldiers preparing to execute a Patriot. In order to stop them, you just need to run or ride your horse into their line of fire. Afterwards you don't have to fight the enemies but continue onwards to the target. You have a limited amount of time.