Projectiles and Moving Frames
Date: Monday, December 11, 2002
I have searched your site for an answer to my question with no luck,
so here goes....
Trajectory question : Please describe the trajectory of a projectile
fired from an airplane under these two circumstances, 1) the
projectile is fired FORWARD, in the direction of travel, and 2) the
projectile is fired REARWARD, directly opposite the direction of
travel. For the sake of argument, let us say the speed of the
aircraft is 1000 feet per second and the speed of the projectile is
also 1000 fps.
I am having difficulty understanding the correlation between the
aircraft, the projectile, with ground speed vs airspeed further
confusing the issue. Does the projectile when fired forward, fly away
from the plane at a rate of 1000 fps.,(with a total rate of travel at
2000 fps. (airspeed...ground speed?)? Conversely when fired away from
the direction of travel, does the projectile just "stay" with the
plane until gravity pulls it to earth?
Let us assume that air speed and ground speed are the same for the
aircraft (i.e., the air is not moving relative to the ground). Since
the projectile is fired from the frame of reference of the aircraft the
projectile velocities are with respect to [wrt] the aircraft speed. The
projectile's ground speed is the aircraft ground speed plus the speed of
the projectile wrt the aircraft. When the projectile is fired forward
the projectile speed is positive when it is fired to the rear its speed
You have the forward case right; the projectile has a ground speed of 2000
fps. It keeps moving at 2000 fps horizontally but its vertical speed
(toward the ground) increases as would any other dropped object. (I am
The reverse case is the same math: the ground speed of the projectile is
zero; it just drops straight to the ground. In both cases the horizontal
speed of the projectile with respect to the plane is 1000 fps.
The plane and the projectile are already moving forward at 1000 fps. The
launch speed is how much the velocity of the projectile changes during the
launch. Let's look at everything as seen from the ground.
When fired forward, the velocity increases from 1000 fps to 2000 fps. Now
the projectile is moving forward much faster than the plane. Assuming air
resistance does not do anything extreme, the projectile will continue with
this horizontal velocity because gravity cannot change it. Now that the
projectile is no longer supported by the plane's floor, it begins to drop.
This is the effect of gravity. The vertical velocity increases the same as
for a dropped ball.
When the projectile is fired backward, the velocity decreases from 1000 fps
to 0 fps. Now the projectile is not moving. The plane continues forward at
1000 fps. The projectile never gets any horizontal velocity because gravity
can't change it. Vertical velocity does change, just as for the
forward-fired projectile. The projectile drops straight down.
In both cases, the difference in velocities between the projectile and the
plane is 1000 fps. To someone on the plane, the forward-fired projectile
looks like it moves at 1000 fps forward. The backward-fired projectile
looks like it moves at 1000 fps backward. You can only say one is more
correct than the other if you know for a fact which observer truly has zero
velocity. This is something nobody knows. Many people swear that the
surface of the Earth does not move. The Earth rotates. At the equator, this
requires a speed of about 25,000 miles per day: more than 1,000 miles per
hour. Just be sure all measurements are in the same reference frame before
applying any laws of physics.
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012