How to solve problems that involve both horizontal and vertical motion
Always keep in mind the following facts:
1.
The vertical component of motion is subjected to gravitational acceleration, therefore
the vertical components of motion are found using the following equations:
A=v
and
d = ½ a t2
t
2.
The horizontal component of motion is not subjected to acceleration so we use the
equation:
V= d
t
3.
The horizontal and vertical components of motion are independent of each other. In
other words, what happens in the vertical direction does not affect what happens in the
horizontal direction.
4.
If an object goes up and then back down, the acceleration changes from negative to
positive at the apex. In this case it, if you are solving for a vertical component of motion,
it is best to divide the “hang time” in half and consider only the upward or downward
portion.
Examples:
a)
A woman in an airplane is traveling horizontally at 80 m/s at an altitude of 600 m. If she
wants to drop a water balloon on her ex-husband, how far in front of him does she need to let go?
Solution: you are looking for a horizontal distance, so you must use v = d / t
You know how fast she is traveling horizontally, but you don’t know the time it will
take to fall.
First find the fall time. This is a vertical component, which involves acceleration. You
know the vertical distance (600 m) and the rate of acceleration (10 m/s/s), so you will
use d = ½ a t2 and solve for time:
600 m = 5 t2
11 s = t
The balloon will continue to move horizontally at 80 m/s (ignoring friction) for the
whole 11 seconds it falls. Therefore we can use v = d / t
80 m/s = d / 11 s
d = 880 m
b)
An arrow is fired vertically at 45 m/s. How high will it go? How long will it be in the air?
Solution: To find how high: the motion is vertical, so acceleration is involved. Height is a
distance, so we will need to use the equation d = ½ a t2 but first we need to find t.
We know two things about the arrow, velocity, and acceleration, so time can be found
using the equation a =  v / t 10 m/s/s = 45 m/s / t
t = 4.5 s
We can now use the time and the acceleration to find the vertical distance (how high)
by using
d = ½ a t2
d = 5 (4.52)
d = 101 m
We can find the time the arrow is in the air by taking the time it took to reach the apex
and doubling it. The arrow is in the air for 4.5 s  2 = 9.0 seconds.
c)
Barney is fired out of a cannon at 185 m/s. If the purple demon leaves the cannon at an
angle of 36 ,how long will he be in the air, how far away will he land?
Solution: The first thing we need to do is to determine the horizontal and vertical components of
his motion:
Vertical component =
Sin (36)  185m/s = 109 m/s
Horizontal component = Cos (36)  185m/s = 150 m/s
The purple miscreant will be airborne for as long as it takes him to go up and come
back down. We therefore are dealing with vertical motion, which is subject to
acceleration.
We are looking for a vertical time, so we will use his vertical velocity (109 m/s)
a=v/t
10m/s/s = 109 m/s / t
t = 10.9 s
Keep in mind that this is just the time he travels upward, his total trip is twice this, so he
is airborne for 10.9 s  2 = 21.8 seconds.
How far away does he land? We are looking for a horizontal distance. So we will use his
Horizontal velocity (150 m/s) in the equation v = d / t and solve for distance, using
his total up/down fall time of 21.8 seconds.
150 m/s = d / 21.8 seconds
d = 3270 meters