Suppose we have a cannon at ground level that shoots a projectile along the path shown. This type of motion is known as projectile motion.

The projectile is fired with a given initial velocity v0, and at a particular angle of elevation a.

The velocity of the projectile at a particular time is given by the vector v.

We wish to determine an equation for the position vector r.

We know that the acceleration vector of the object points downward, and its magnitude is given by the acceleration due to gravity g.

After determining the velocity vector, we can solve for the vector constant of integration using the initial condition for the velocity.

We then substitute for the constant of integration in the velocity vector:

We now have the position vector. We can solve for the vector constant of integration using the initial condition for the position. The projectile is at the origin at time t equals zero.

Substituting for the constant of integration produces an expression for the position vector.

We can express the initial velocity vector as x and y components.

This gives us the final expression for the position vector for projectile motion.

The horizontal distance traveled by the projectile is known as the range. Use the slider to vary the angle of elevation. Then, press the button in order to shoot the cannon. Notice how the range of the projectile varies with angle of elevation.

Consider a projectile with the given initial speed.

What is the velocity vector of the projectile? Click the "Submit" button after selecting your answer.

Now, we can determine the position vector. What is the position vector of the projectile? Click the "Submit" button after selecting your answer.

At what angle of elevation is the range of the projectile a maximum? Click the "Submit" button after selecting your answer.

If the angle of elevation of the cannon is set for maximum range, what is the position of the projectile at t equals two? Click the "Submit" button after selecting your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education