How to Calculate Force on a Projectile

Newton's cradle-a demonstration of Newton's second and third laws

The force on a projectile after its initial acceleration includes gravitational force and air resistance. The gravitational force points vertically downward. The air resistance points opposite to the direction of motion. In the case of free fall, a projectile can reach a maximum velocity, or "terminal velocity," past which it won't accelerate because the gravitational and air resistance forces balance out. In such a case, the net force on the projectile is zero.

Instructions

1. Recall Newton's second law, that force equals mass times acceleration, or F=ma. Then the gravitational force on a projectile is F=mg, where g is the gravitational acceleration constant, which is approximately 9.80 meters per second-squared. If m is in kilograms, then F is in Newtons.

2. Denote the velocity vector as V. Then the frictional force on a projectile as -C1 V -C2 V mag(V), where mag(V) is the magnitude of the vector V, i.e. the speed. (Remember, V is a vector, with a directional component; therefore, squaring V wouldn't have made sense. The velocity direction needs to be retained.) Here, C1 and C2 are constants that can be determined empirically. For most speeds, the second term dominates because it grows geometrically instead of linearly.

3. Determine the resultant force, or combination of the two forces in Steps 1 and 2, by using vector addition. In other words, attach the head of the friction force's vector onto the tail of the gravitational force vector without changing either vector's orientation. Then draw a vector from the tail of the friction vector to the head of the gravitational vector. This is the total force exerted on the projectile. Of course, when the projectile is in free fall at terminal velocity, the friction vector and gravitational vector are equal and opposite in direction, so the vector addition produces a resultant vector of zero.

Tips & Warnings

• For objects with low air resistance, as opposed to a falling feather, C2 is easier to determine than C1. If you can determine the object's terminal velocity, then you can find C2. For relevant formulas, see, for example, Fowles' "Analytical Mechanics," shown in the References section.

References

• "Fundamentals of Physics"; David Halliday and Robert Resnick; 1992
• "Analytical Mechanics"; Grant Fowles; 1986
• Georgia State University: Projectile Motion