Make - это универсальный инструмент, и в нем много разных тонкостей. Тем не менее, вы можете сделать все просто:
OBJ := ai/utils/geometry.o ai/world/world.o ai/world/ball.o ai/world/bat.o
all: ai/client
.PHONY: all # specific to GNU make, which is what Ubuntu provides
ai/client: ai/client.o $OBJ
# this rule means each .cpp file depends on its corresponding header
# and, since the .o files depend on .cpp files (a builtin make rule),
# they will be recompiled if the headers change
#
# you can also get more complex and generate dependencies automatically
# look at the -MM option for gcc, for example
%.cpp: %.h