## Horizontal Projectile Motion Math

### Create a Separate X and Y Givens List

Horizontal projectile motion math problems start with an object in the air beginning with only horizontal velocity.  These problems often start with an object rolled off a table, being thrown horizontally, or dropped by a plane.

When we approach horizontal projectile motion problems separating the x and y components and work through problems keeping those givens separate.

When you see a problem with an object in the air traveling going forward but under the influence of gravity, create a separate X and Y givens list. ### Projectile Motion Equations

Look at the equations used in projectile motion below.  In the X axis you will only use our constant motion equation. Below you will see vx which is just velocity in the x axis.

In the Y axis you will use our common acceleration equations.  Below they are just specialized for something in the air.  Our normal variable a (acceleration) is exchanged for g (acceleration due to gravity).   X is exchanged for since the object will be moving in the Y axis.  Also the vi and vf are replaced with viy and vfy just representing that the velocities are only Y axis components. Horizontal projectile motion problems will have separate X and Y givens with only time being in both givens list.  Projectile motion problems end when the object hits the ground.  At that moment the projectile is done going forward and falling.

### Two ways to find time:

• If you have the Y displacement you can find time using Y axis givens.  Don't forget that viy = 0 m/s and g = 0 m/s2
• If you have horizontal velocity (vx) and X axis displacement (X), you can find time in this axis

### Example:

Joe throws a ball 9.4 m/s horizontally forward from the second floor 6 meters off the ground, how far downrange does it land?

What you see (in the red to the right) are givens in all horizontal projectile motion problems.  All falling from rest and accelerate at 9.8 m/s2 down.  Since all the directions in the Y axis are down well call down the positive direction in the math.

We have three of four givens for the normal acceleration equations and since Viy = 0 m/s you can use the equation you see on the right in the picture.  Note that this is the rearranged version of x = vit + ½ at2 in the 1D motion equations.  If you are not provided the simplified equation you'd start there. When you take the square root of 2 times 6 divided by 9.8 you get 1.107 seconds.  This is the time it takes to fall.  This is also the time it has to go forward so throw it in the x axis givens list. Lastly you use the constant motion equation in the X axis to solve for x.

• x = (vx)(t)
• x = (9.4)(1.107) = 10.41 m forward

### Example:

A ball rolls off a table at 3.0 m/s and lands 2.5 meters away.  How high was the table?

We have two of three X givens and solve for time using the X axis and the constant velocity equation seen on in the picture. • t = x/vx
• t = 2.5/3.0 = 0.833 s
• Place time in the Y axis givens list and solve for y
•  y = viyt + ½ gt2
• y = (0)(0.833) + (0.5)(9.8)(0.8332)
• y = 3.4 m down

### Horizontal Projectile Motion Math Quiz

Horizontal Projectile Motion Problem Quiz

Do you know how to solve a horizontal projectile motion problem? 1 / 4

A ball is kicked horizontally from a 45 meter tall cliff at 20 m/s.  How far from the base of the cliff does it land?

2 / 4

A drone looses battery power from a height of 72 meter while flying horizontally at 12 m/s.  How far from the ground below will it land forward? (neglecting air resistance)

3 / 4

A bullet shot horizontally at 1400 m/s lands 300 meters downrange.  From what height was is shot from?

4 / 4

A bullet shot horizontally at 1400 m/s lands 300 meters downrange.  From what height was is shot from?