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Do heavier bullets drop faster than lighter bullets....

Discussion in 'General Gun Discussions' started by GarandOwner, Jul 9, 2008.

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  1. GarandOwner

    GarandOwner Well-Known Member

    One great gun debate that has been brought up recently is whether bullets of a heavier weight drop faster than a lighter bullet of the same size. Many refer back to basic physics that states that in a vacuum two objects that are acted on by the same acceleration (in this case gravity) will move at the same rate regardless of mass or size. It is this concept that has led many to claim that bullets drop at the same rate regardless of their weight. However there is a key component missing in this claim. In the real world, we don't shoot in vacuums, we shoot through air. For this reason we must account for air resistance. Air resistance occurs from the flow of air around an object as it moves. In the case of bullet drop, there are two components of air resistance, a vertical resistance and a horizontal resistance. The only one that has an impact on the amount of time it takes for a bullet to hit the ground is its vertical resistance. One concept that carries over regardless of wither or not the bullet is moving through a vacuum or a fluid, is that a bullet that is shot parallel to the ground will hit the ground at the exact same moment as one that is dropped at the same time. For this reason we can examine free fall to determine if an objects weight is significant in the time it takes to hit the ground. It does use some higher level math (calculus and differential equations) but I tried to explain it simple enough that anyone can follow and understand.

    First case: Neglecting air resistance

    position, velocity and acceleration are all related to one another. Velocity is the derivative of position, and acceleration is the derivative of velocity, and the second derivative of position. First we will start by looking at acceleration. A bullet is generally fired no more than 6 feet above the ground, for this small variation we can neglect the changes in acceleration due to gravity since it is minimal (minimal meaning that it doesn't even change out to the one ten thousandth place holder. (.0001)) so acceleration is a constant g that is not dependent on time. As with all physics equations, we start with Newton's second law: F = ma where a force equals a mass times its acceleration. We know gravity, but we want to find it as a function of time. So we have:

    ma(t) = mg

    where m is mass, a(t) is the acceleration as a function of time, and g is gravity

    We can see that mass cancels from both sides and we are left with:

    a(t) = g

    Taking the anti derivative (Integral) of acceleration we can get the velocity function. Since gravity is a constant it is a simple integral that evaluates as:

    v(t) = gt + C1

    Once again we can take the anti derivative to find the position as a function of time.

    h(t) = 1/2gt^2 + C1t + C2

    If we impose the initial conditions v(0) = Vo and h(0) = ho then this equation becomes:

    h(t) = 1/2gt^2 + Vot + ho Where Vo is the initial velocity and ho is the initial position.

    This function shows that an objects position is independent of its mass, so no matter the mass of the object, it will hit the ground at the exact same time. However this function has a flaw: it neglects air resistance, as we all know air resistance is significant. So let us start again including air resistance.

    Second Case: bullet drop with air resistance

    Air resistance is bv, where b is a constant that depends on the objects shape and the density of air, and v is the objects vertical velocity. Once again we get our basis for the equation from F = ma so this time we have:

    ma(t) = mg - bv (air resistance is negative because it is an acceleration that "slows" the effect of gravity)

    Once again to find velocity we must integrate with respect to time: It must be noted that this time our acceleration function has a velocity component, for this reason we can make it easier on ourseves by writing acceleration as the derivative of velocity with respect to time (dv/dt) so we have:

    m (dv/dt) = mg - bv

    by rearranging the equation we can make it more manageable to integrate

    dv/(mg - bv) = dt/m

    now we integrate each side, the left side is integrated with respect to velocity, and the right side with respect to time. So the equation becomes:

    -(1/b)ln|mg - bv| = t/b +c where c is the integrating constant

    we want to isolate velocity by itself so we multiply both sides by -b and take the exponential function of each side to help break down the left side this gives us:

    mg - bv = e^(-bc)*e^(-bt/m)

    which can be written as:

    mg - bv = Ce^(-bt/m) where C = e^(-bc)

    some simple algebra re-aranges the function so we have v by itself, this is:

    v = (mg/b) - (C/b)e^(-bt/m)

    by once again imposing the initial condition where v(0) = Vo we can solve for C. We see that when time is zero the exponetial term goes to 1 so:

    Vo = (mg/b) - (C/b)*1

    solving for C here we get that C = (Vo)b - mg plugging this back into our velocity equation yields:

    v = (mg/b) + (Vo - (mg/b))e^(-bt/m)

    It can be seen that velocity is a function that IS Dependant on mass when air resistance is included, so this means that the weight of a bullet DOES effect how fast it drops. Plugging in some simple values will show that an object of the same shape (same b value) with a larger mass WILL in fact drop FASTER than one that is the same shape but lighter.

    For those over zealous mathematicians/physicists/engineers out there, you can integrate again and get the position function.
  2. razorblade31

    razorblade31 Active Member

    True as far as it goes, but the fact is a heavier bullet is probably going to be a different shape than a lighter one. So the calculations you show cannot actually tell us if a heavier bullet will drop faster unless you come up with a satisfactory equation for b.
  3. yeti

    yeti Well-Known Member

  4. ClickClickD'oh

    ClickClickD'oh Well-Known Member

    Short answer: Possibly yes, but not to any significant extend considering the actual distance bullets drop in flight. Forward velocity of a bullet is much more important in determining bullet drop over range than vertical velocity is.
  5. BammaYankee

    BammaYankee Well-Known Member

    Yeahh uhhh... Here's a thought... Put down the calculator, pick up a rifle, shoot different loads at 1000 yards, and see for yourself.
  6. freakshow10mm

    freakshow10mm member

    No, lighter bullets drop faster. They lose velocity quicker than heavier bullets.

    MT GUNNY Well-Known Member

  8. GigaBuist

    GigaBuist Well-Known Member

    Well, yes, but that doesn't have any appreciable meaning until you start reaching vertical speeds approaching the terminal velocity for the projectiles involved.
  9. GarandOwner

    GarandOwner Well-Known Member

    I cant, it rained today :neener:

    The velocity in this equation is the vertical velocity (down) not horizontal velocity
  10. freakshow10mm

    freakshow10mm member

    I don't speak math. I speak experience.
  11. kcshooter

    kcshooter Well-Known Member

    If you hold a bullet exactly the same as the one in the round loaded in your chamber at the same height as your chamber and drop it at the same time as the gun fires, the round shot from the barrel hits the ground at the same time as the one you dropped if your barrel parallel to the ground.
    So the real test for this would be dropping a 55gr bullet and a 230gr bullet at the same time and seeing which one hits the ground first.
  12. yeti

    yeti Well-Known Member

    Well that one I can answer, they both hit at the same time.
  13. PercyShelley

    PercyShelley Well-Known Member

    Lighter bullets would "drop faster" in the same, intuitive, trajectory-related sense that lower velocity bullets "drop faster", this is, they have more drop.

    Also, "heavier bullets", at least within a given caliber is usually taken to mean longer bullets of similar density to the lighter bullets. Unless I'm missing something, these calculations seem to apply to a case where only the bullet's mass changed; which in the absence of a change of form means a change in density.

    A longer bullet might well have a lower ballistic coefficient than a shorter one when measured sideways.
  14. Seancass

    Seancass Well-Known Member

    Same speed, more mass means more energy. Since it has more energy, it will be less effected by the air resistance and will maintain it's high speed longer. (mass @ velocity, given air resistance produces deceleration)

    Your math does seem to prove that a heavier bullet will drop faster, but energy says that the heavier bullet will also have a higher speed at a given range. So, does the substancial increase in energy over come the minute increase in downward aceleration?

    I think a heavier bullet will actually have enough energy to maintain it's high velocity longer and actually appear to fall slower than a lighter bullet.
  15. Benzene

    Benzene Well-Known Member


    BammaYankee, I fully agree with your method to "pick up a rifle, shoot different loads at 1000 yards, and see for yourself."

    A rigorous mathematical analysis might be making too many assumptions. For example, how could it be 100% certain that deceleration is directly proportional to speed and not to speed-squared? What about turbulence factors of the medium (air)? And at these speeds and for a significant segment of the shooting population, is there a significant difference between likely drops?

    Nice mathematical analysis, though.
  16. RedLion

    RedLion Well-Known Member

    Dang it! You made me have to open my physics book and it is still SUMMER!!! but you are correct. Fluid resistance does change how stuff falls on the earth, thats why the astronauts bothered bringing a bowling ball and a feather to the moon and dropping them so to prove galileo's theory.

    If there are any doubts, try dropping the bullets *sideways* in a fish bowl or something with water so the difference is more noticeable.
  17. misANTHrope

    misANTHrope Well-Known Member

    Even if both bullets dropped vertically at identical speeds, given equal powder loads, the lighter bullet would drop less over a given distance due to increased horizontal velocity. Of course, that becomes less true the further downrange you get, because then you're moving into the realm of which bullet will retain velocity better... and I don't feel like trying to remember how to derive some real numbers there.

    We can say for certain that equal forces of air resistance will decelerate a heavier object less. On the other hand, heavier bullets sometimes have different cross-sections from lighter- observe the difference between typical Core-Lokt .30-06 in 180 and 150 grain loads. The heavier load has a round nose. As for how much of an effect on drag that has? Beyond my ability to figure out with any kind of reliability.

    In the end, modeling using mathematics will always involve assumptions of some kind, and will never be a substitute for real-world testing. But it's fun to think about. :)
  18. justin 561

    justin 561 Well-Known Member

    Did they change the definition of vertical? I always seem to have thought it meant Up and down.

    Am I the only one confused? Did Newtons law of Gravity change? No matter how heavy a object is, it's going to fall at the same speed regardless of weight when vertically dropped. If you mistyped and meant to type "Horizontal" then you're probably right.
  19. Jimmy Dean

    Jimmy Dean Well-Known Member

    Umm, Mis, horizontal velocity has literally NO effect on bullet drop, in essance because the bullet itself produces zero list capability. IF the bullet produced lift, then horizontal velocity and vertical velocity would be related, but since it does not, they are not.

    In relation to TIME, all bullets fired exactly horizontal will drop at exactly the same speed (this is for modern bullets, not round-ball bullets which can actually produce lift-like aerodynamic properties)

    Now, over distance, differant rounds/loads will drop at a differant rate per amount of feet traveled foward, this is simply due to the fact that one bullet may travel 1500 feet in 2 seconds, whereas another bullet only travels 1000 in 2 seconds, and yet a third will travel 2000 feet.

    At the 2 second mark, all bullets have dropped the same, but at the 1,000 foot mark, their drops are differant, because for the fastest bullet, that is only 1 second into flight, for the slowest, that is 2 seconds, it will have dropped signifigantly further by that distance.
  20. gallo

    gallo Well-Known Member


    It's been more than 8 years since I took a calculus class. From what I can understand, yes the weight of a bullet affects how fast it drops. However, two bullets of the same shape can be accelerated differently, as in the case of .357 magnum and a .38, and therefore drop at different rates.
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