**Projectile motion** is the motion of an object which has no force acting on it but the gravitational field. This type of object is called a *projectile*.

Watch this video for an excellent overview of what a projectile motion is (with the NFL and Lester Holt):

## Projectile Motion Without Initial Velocity

In the simplest kind of projectile motion problems, there is no initial velocity. An object is simply dropped so that the Earth’s magnetic field pulls it toward the ground at a rate of 9.81 m/s^{2}.This acceleration is all in a vertical direction (i.e. toward the Earth’s surface). Gravity doesn’t have a horizontal component, and so (unlike the football in the above video), there isn’t any horizontal component to this type of acceleration: the object simply drops straight down.

For example, if you stuck your hand out of a skyscraper window and dropped an apple, it would have a velocity of 0 at the moment you opened your hand (t_{0}). At t = 1 second, the apple would be traveling at 9.81 m/s^{2}. At *k* seconds in, it would be traveling downward at a speed of 9.81m/s^{2}*k*, supposing it hadn’t hit the ground yet and assuming (for simplicity) that we can ignore drag.

## Projectile Motion With Initial Velocity

Many times, a projectile will begin with an initial velocity. We defined a projectile as an object which has no force acting on it in the present except the gravitational field. It’s important to realize, though, that it may have had other forces acting on it in the past. Your hand, for instance, may have been exerting force on the ball as you threw it out.

Suppose you are standing by the window in your building, 150 meters above the earth, and you throw your ball at an angle of 20 degrees above the horizontal, with an initial speed of 8.40 m/s.

The moment the ball leaves your hand, at t_{0}, it is traveling at 8.40 m/s, and from that point forward the only force acting on it is gravity, which gives it an acceleration of 9.81 m/s^{2}. The ball begins by going up, but one second later, it will already have begun moving down. You can’t add or subtract these rates directly; they are vector quantities, which means we have to pay attention to direction and add and subtract only what matches.

Let’s say that you are standing at the origin of a simple x y Cartesian coordinate system. When you throw that ball out at a 20 degree angle, its velocity will have two parts. Call them v_{0x} and v_{0y}. Draw a triangle, and use your geometry skills to solve for v_{0x} and v_{0y}.

- v
_{0x}= v_{0x}cos (20) = 8.40 cos(20) = 7.893 m/s, and - v
_{0y}= v_{0y}sin (20) = 8.40 sin 20 = 2.873.

Since we’re pretending the effect of drag is negligible, there is no force acting on our ball horizontally: a_{x} = 0. v_{0x} = v_{x} = 7.893 m/s at every point until the ball crashes to the ground.

It’s the vertical velocity that will be acted on by gravity. At t_{0} that was 2.873, in a + or upward direction, and gravity causes it to accelerate at 9.81 m/s^{2} in a – or downward direction. One minute in, it’s vertical speed will be 2.873 – 9.81 = -6.937, or about 6.9 m/s in a downward direction.

## References

Allain, Rhett. Projectile Motion. Introductory Physics Lab. Retrieved from https://www2.southeastern.edu/Academics/Faculty/rallain/plab193/page1/page31/page31.html on January 26, 2019

Duffy, Andrew. Projectile Motion Example. Retrieved from http://physics.bu.edu/~duffy/semester1/c3_projectile.html on January 26, 2019

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