A projectile is any object moving through a space under its own inertia and the only force acting upon the object is gravity. Examples of projectiles include various sports balls thrown through the air and bullets (after it leaves the weapon). Objects such as rockets, airplanes and flying birds are not considered projectiles because they provide their own propulsion force to keep them in motion. The projectile’s motion is generally the path (or trajectory) taken by the projectile as it moves through the air. When studying projectile motion we are mostly conserned with motion near the earth’s surface where the force exerted by gravity can be concerned constant with time and space.

Projectiles With No Drag

Let us consider the simplest case; projectile motion with no air friction in two dimensions. Assume a cannon fires a cannon ball at some angle. Then in vector notation. Newton’s second law becomes.

Where r is a vector signifiying the position of the particle and g represents the acceleration of gravity as a vector. This can also be written in unit vector notation:

Here, i is a unit vector in the positive x direction, and j is a unit vector in the positive y direction. The right side of the equation is negative because gravity pulls objects towards the earth. We can now write this differential equation and two separate equations.

Equation 3 and 4 are independant separable separable differentials that have already been solved in the Equations of Motion applet. The solutions to the equations are:

If we position our coordinate system such that the canon is at (0,0), x0 and y0 are 0. The component vx0 and vy0 are the x and y components of the initial velocity when it leaves the cannon. If we know the angle that the cannon fires the projectile at, we can rewrite the components as:

Here, v is the magnitude of the velocity of the projectile when it leaves the cannon, and the angle is the angle the cannon fires the projectile at.

Flight Time, Maximum Distance and Maximum Height.

The flight time of the projectile is the time the projectile is in the air. This can be determined quite easily. If a projectile is shot out of a cannon at some angle, the projectile will increase in height, and then fall back to the ground where it reaches its original y-displacement of 0. So using equation 8, and setting y(t) = 0, we can solve for the time at which the projectile has reached a displacement of 0. Doing so yields the flight time expression:

Substituting equation 11 into equation 6, gives us the maximum distance the projectile can travel, or the range of the projectile:

Now consider the motion in the y-direction. The projectile gets fired and moves vertically until it reaches a maximum height, then starts to fall back down towards earth. At the maximum height, the y component of the velocity is zero for a split second. Using equation 7, we can solve for when the velocity is zero and substitute that quanity into equation 8 to determine the maximum height. We can also find the maximum height by noticing that the motion of the particle in the y-direction is a parabolic function and therefore symmetric, therefore at half the flight time, the particle would be at it’s maximum height. This would only work if we consider the fact that there is no air resistance. This method will not work if there is drag. So instead, let us use the first method. Doing so yields the following expression:

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