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Trajectory: Definition & Equation

Instructor: Scott van Tonningen

Scott has a Ph.D. in electrical engineering and has taught a variety of college-level engineering, math and science courses.

In this lesson, we'll explore the science behind ballistic trajectories and the equations for doing some simple calculations. A quiz is provided to help test your knowledge.

Skyrockets in Flight

There is nothing, in my mind, like a good fireworks display. For centuries, people have launched rockets into the air with the express purpose of blowing them up in a spectacular way. The last time I saw a fireworks display, I noticed that the timing of the explosions seemed to fall into three categories: 1) those that exploded as the rocket was on the way up; 2) those that exploded at the top of the rocket's path; and 3) those that exploded as the rocket was on the way down. I also noticed that not all of the rockets were fired straight up; some were fired at an angle in one direction or another.

Fireworks display at night
Fireworks display

I did some reading on how fireworks are constructed, and I discovered that in addition to the fast-burning fuse used to launch the rocket from the ground, there is another slow-burning fuse inside the rocket that is timed to create the explosion at just the right moment during flight. How do the folks that build these skyrockets know what that right moment is? Suppose they want it to explode at exactly the top of the trajectory. In addition, suppose they want to point the launcher at such an angle that the explosion occurs not only at the top of the trajectory but also a certain distance away from the launcher.

Look at the following diagram. A skyrocket is launched from the ground to a height of 250 feet in about 5 seconds. Let's say we want the rocket to explode exactly at the top of the trajectory. If the rocket explodes at 3-4 seconds, it will be too early. But 6-7 seconds would be too late. Also, this rocket is launched at a predetermined angle from the ground so that it will travel downrange a predictable distance during that 5 seconds. If the angle is too close to vertical, then the fireworks people won't get the desired effect. If the angle is too close to horizontal, then the rocket will travel too far downrange. How do they know what angle to use?

Diagram of ideal skyrocket timing
Ideal fireworks rocket timing

Fortunately for the fireworks manufacturers, the physics behind calculating a trajectory are very well understood. In fact, it is these calculations that are used in all kinds of interesting applications, such as ballistics, planning missile launches, hitting baseballs and designing golf clubs.

What is a Trajectory?

A trajectory is nothing more than the path an object follows as it moves through space. For this lesson, we are going to concentrate on a ballistic trajectory, which is the path an object follows through space after it is initially launched, with its path dictated only by the laws of motion, gravity and possibly air resistance.

How to Calculate a Ballistic Trajectory

The easiest way to do ballistic trajectory calculations is to separate the movement of the object into two independent parts, the purely horizontal motion and the purely vertical motion. It turns out that these two motions can be calculated independently and then the results are combined back together in a meaningful way. For all of these calculations, we will assume that the object is launched from the ground and that it will eventually hit the ground again at the same elevation.

Let's look at the horizontal direction first. Assuming no air resistance and no gravity, an object launched horizontally with a certain speed will continue moving in the same direction, at the same speed, forever. If we assume the air resistance is negligible (close to zero), this provides us with the equation we need for the horizontal trajectory component. It is very simple: distance is equal to the horizontal speed multiplied by time. The problem is that the object is usually not launched in a purely horizontal direction. The launcher will have some angle with respect to horizontal. We need to know what that angle is so we can figure out what proportion of the actual launch velocity is in the horizontal direction. Look at the diagram.

Diagram of horizontal and vertical trajectory components
Horizontal and vertical trajectory components

Trigonometry is very handy in this situation. The horizontal component is defined by using the cosine of the angle shown on the diagram. So, if the object is launched with a certain initial velocity, Vo, there are two simple equations that result.

Horizontal movement equations

Now let's tackle the vertical component. Again, we will assume zero air resistance, but now we have to deal with gravity. Gravity causes a downward constant acceleration force on the object that we have to consider throughout its flight. If we are close to the Earth's surface, that acceleration value is about 32 feet per second squared. The bottom line is that vertical velocity of the object is not constant at all. It starts out as the vertical component of the initial velocity, slows until the object reaches the maximum height, hits zero for a moment, and then starts increasing in the opposite direction as the object free-falls. The vertical component is defined by using the sine of the angle.

There are several well-known equations for vertical motion that result from this analysis.

Vertical movement equations

There are two other useful equations that are found by combining some of the equations above with the horizontal motion equations.

Additional horizontal movement equations

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