Begin by drawing a picture, showing the spring and dart. Define the coordinate system and indicate the compression of the spring.
Figure: A compressed spring and dart in a loaded dart gun.
(a) At the instant the spring is released, it is compressed 0.04 m from its relaxed length. A freebody diagram for the dart at this instant is shown below.
Figure: A freebody diagram for the dart when the spring is fully compressed.
The net force on the dart is the sum of the forces on the dart. The net force on the dart can be found by adding the vectors tail to head from the freebody diagram, and by summing the forces algebraically. The algebraic answer should be consistent with the picture, as shown below.
Figure: A freebody diagram for the dart.
The spring force is given by Hooke's law:
In this case, the spring has a stiffness and is compressed 0.04 m from its relaxed length and is directed upward. Substitute the spring force and gravitational force on the dart and solve for the net force.
The net force is upward which is consistent with our freebody diagram and our picture showing the sum of the forces on the dart.
(b) At the instant the dart leaves the spring, the spring is at its relaxed length. In this case, the only force acting on the dart is the gravitational force on the dart.
A picture of the dart and a freebody diagram at the instant the dart leaves the spring are shown below. The freebody diagram for the dart is consistent with our calculation.
Figure: A freebody diagram for the dart when it leaves the spring.
