Is Acceleration due to gravity independent of horizontal velocity?

Question

Is it true that all objects, irrespective of mass, reach the ground at the same time when dropped from the same height at the same time ?

If yes, do the following objects reach the ground at the same time

a) a bullet fired horizontally from a height x above the ground
b) a bullet dropped vertically (free fall) from the same height x above the ground?

Answer 1

In a vacuum, all of the above would be true, and are true.

However, I believe that, with an atmosphere thrown into the mix, as you will, invariably, find in the real world, you have principles of aerodynamics to contend with, where things like lift (often induced from thrust alone) counter gravity's effect on an object's motion.

Go exploring through these categories for more definitive and thorough explanations:

Answer 2

The answer to your first question is YES, all objects, irrespective of mass, will reach the ground at the same time when dropped from the same height at the same time. But we must assume that there is NO air resistance, i.e. nothing else in the vertical direction that would counteract the uniform acceleration due to gravity.

So the answer to your second question is also YES given the same assumption that there is NO air resistance, and that the bullet fired from the gun travels a distance that is comparatively short relative to the curvature of the Earth (I think this is a VERY safe assumption. I don't know what the hell the previous responder is talking about). Both bullets will reach the ground at exactly the SAME time. This is because the equation of motion for both bullets are independent of their mass. See below:

Bullet 1 (fired from a gun horizontally):

x = v(x)t + 1/2 a(x)t^2, v(x) is velocity in the x direction and
a(x) is acceleration in the x direction
y = v(y)t + 1/2 gt^2, v(y) is velocity in the y direction and
g is acceleration due to gravity

Bullet 2 (dropped to the ground):

x = 0, no motion in the horizontal direction
y = v(y)t + 1/2gt^2, notice here that the equation is the
same as bullet 1

For the sake of simplicity, let's assume that bullet 1 is fired from the gun with a constant velocity V from a height of H and that there is no acceleration in the horizontal direction. Then

Bullet 1:

x = Vt, notice here that given NO air resistance, the bullet can
travel in the horizontal direction indefinitely
H = 1/2 gt^2, v(y) is zero because it has NO initial velocity in the
y direction

this implies that the bullet travels in the vertical direction for a period of time, t = sqrt(2H/g). This then means that bullet 1 travels in the X direction a distance of X = sqrt(2H/g)V.

Notice that for bullet 2, which is dropped from height H, it also falls and reached the ground at t = sqrt(2H/g). The only difference between bullet 1 and 2 is that bullet 1 travelled along the X direction for a distance of sqrt(2H/g)V because it was fired from a gun. And because there is NO air resistance, bullet 1 just keeps moving in the X direction until gravity pulls it down and it hits the ground which causes it to stop.

Answer 3

acceleration from gravity is a constant, while the velocity is the speed the object is traveling at. All objects regardless of mass will reach the graound at the same time as long as the following are applicable:
1. wind resistance for both is identical (a piece of paper has more wind resistance than a bowling ball
2.the ammount of lift generated by the objects is the same (a paper airplane will hit the ground later because when it reaches a certain speed it levels off)
and 3. the forces reacting on the objects are identical (if there is wind pushing one object down and not the other the one will fall faster.

For the bullet question, the bullets would land in the same time frame, but bullet a will travel very far forward in that time. neither would have a downward velocity at start and so their downward acceleration and their downward velocity when dropped or fired would be the same. and would thus fall the same distance vertically in the same ammount of time although total distance traveled including horizontal would be different. Hope that solves the problem.

Answer 4

The answer to the first question is yes providing that the mass of each object is very small compared with the mass of the earth. Remember that gravity is a mutual attraction, that is to say that the earth and an object fall towards each other. To be really silly, the earth and Jupiter would move towards each other with a greater acceleration than the earth and a feather.

The answer to the second is of course not. By horizontally I hope you mean parallel to a tangent to the earth's surface. This being the case, without the influence of gravity the bullet fired from a gun would move away from the earth's surface because the earth is curved. So, we start with one bullet that is not moving away from the surface and one that is. The fired bullet will hit the surface after the other one.

Answer 5

When a bullet is dropped down it is pulled by the earth and its speed increases as it falls down; or the speed increases as time of travel increases.

If the bullet is fired horizontally, the earth does not stop its pulling action. Therefore as in the above case, it is pulled down and its speed in the vertically downward direction increases as time of travel increases. At the same time it is moving horizontally with a speed because of the firing.

Therefore, the bullet falls down as well as moves horizontally. The combined path is a parabolic path and the time of touching the ground is the time it will take when dropped.

The time for the bullet to reach the ground will be small, if dropped down instead of firing. However, if fired from that height horizontally, it covers longer horizontal distance during that small time because of its high speed.

Answer 6

YES BUT they will not hit at the same time!

If you fired a bullet horizontally (i.e., perpendicular to the line connecting the center of gravity and you) then you are actually firing it away from the center of gravity i.e., in a short time it will be further away.

Its easy to think of it on a big scale. Like make yourself really big relative to the earth where you can see the curvature. If you put your arms out... the tip of your arms is actually further away from the center of the earth (the center of gravity) Thus since when you get further away from the center of gravity the weaker the gravitational pull and also the further you have to drop.

Remember while the earth seems flat it is actually curved.

Remember also this only applies in a vacuum with a perfectly smooth surface and sphere. Otherwise the previous answer about the unpredictability is the most accurate.

Answer 7

The scary part is the number of people who said no.

In a vacuum, yes they hit the ground at the same time.

In the real world, probably not. Aerodynamics (except for drag) plays a very small part (usually not even measurable) in the flight of a bullet.

What *does* count big-time is any offset between the longitudinal axis of the spin imparted by the lands and grooves in the barrel, and the longitudinal axis of the true center of mass of the bullet. When the bullet is mechanically constrained to spin in the barrel, it does so. But when it leaves the muzzle, it will *immediately* begin to spin around its longitudinal center of mass. If the two aren't the same, there will be a lateral force applied to the bullet (in a completely unpredictable direction) so it may have a vertical component to it's motion when it starts its trajectory.

One of the little things you learn after you've been loading your own ammo since you were 10 and been around guns and shooting all 61 years of your life


Doug

Answer 8

Perpendicular vectors are independent.

Over an infinite plain, two objects dropped with identical vertical velocities in a uniform gravitational field in a vacuum will hit the ground at the exact same time regardless of mass or horizontal velocity.

In other words, yes.
We can, for the purposes of this example, ignore the curvature of the Earth and air resistance. Gravity will be accelerating both object's equally towards the ground, they will hit at the same time if both we dropped at the same height.

Answer 9

Yes and No. The bullet fired horizontally will have some horizontal force on it and hence it would go horizontal first before going vertically down.

Gravity is independent of horizontal velocity only in true and perfect condition. Hence, in vacuum. So if you drop a feather and bullet from the same height in vacuum. They will hit the ground at the same time.

Physics 101

Answer 10

Ignoring stuff like wind and air resistance, objects released at the same height at the same time, neither of which has any vertical force on them, would fall at the same rate. However, they may not hit the ground at the same time because they'll hit the ground at two different places and the ground may be of different heights.

In particular, (b) would probably hit the ground before (a) because the earth is round and would curve away from (a) so (a) would have a bit further to fall before hitting the ground.

Answer 11

Yes,

No the bullets would not hit the ground at the same time because of what we just said yes to and that is that Acceleration due to gravity is independent of Horizontal Velocity. That's why a Rocket can leave earth's atmosphere and a Turbine Jet cannot.