Calculate the Distance Traveled by an Object Using Vector Kinematics

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Have you ever wondered how far a bullet would travel if you fired it? This can be calculated by finding the angle of launch and the bullet’s initial velocity. Read on to find out how.

Steps

  1. Determine the angle which the bullet is fired, relative to the ground. This can be done using a simple mathematical compass.
  2. Determine the initial (starting) velocity of the bullet as soon as it leaves the barrel of the gun. This information can usually be found on the gun’s package or the manufacturer's website. The initial velocity of the projectile varies, depending on the gun model and ammunition type.
  3. With a writing utensil and paper, construct an x-y coordinate system with the origin located at the point where the bullet leaves the barrel.
  4. Using trigonometry, determine the initial velocity in the x/y-direction. This can be done using the sine and cosine functions. For example, if the angle of launch is 30o, then the sine of 30o, multiplied by the initial velocity, would be the initial velocity in the y-direction. This can easily be done by using this trigonometric function on a calculator: sin(30) * vo. Then, repeat this step, using the cosine function to determine the initial velocity in the x-direction.
  5. Using the equation y= yo + voy – ½ gt2, calculate the time it takes for the bullet to reach the ground. The definition of each variable is as follows: y=final (y-direction) position, yo=starting (y-direction position), voy=initial velocity in the y-direction (calculated in step 4.), g=force of gravity acting on the projectile (assuming we are on earth, this value should be (-9.80m/s2), and t=the time it takes for the projectile to hit the ground. To make this step simpler, solve for (t) using the following equation: t=((y – yo – voy)/.5g)1/2.
    • For example, assuming the final y-position was zero (because the bullet will lose momentum and hit the ground), the initial y-position was 1.5m, the initial y-velocity was 20m/s, and the force of gravity (g) was -9.80m/s2, it should've taken 13.6 seconds for the bullet to hit the ground.
  6. Calculate the distance travelled by the bullet. Using the equation x = vxo t, supply the missing variables vxo=initial (x-direction) velocity, calculated in step 4, t=time for projectile to travel (calculated in step 5). For example, if the initial x-velocity was 20m/s and the bullet's airtime was 15 seconds, the horizontal distance that the bullet travelled was {{safesubst:#invoke:convert|convert}}.

Warnings

  • This Wiki-How article assumes that the bullet or object in motion is only being affected by the force of gravity on earth as well as initial force impacted upon the object.
  • Improper use of weaponry or projectiles can result in injury or death; these calculations can be done without the launching/firing of a projectile simply by supplying the missing variables.
  • This Wiki-How article assumes that the object is in simple parabolic motion.

Things You'll Need

  • Calculator, projectile (bullet, football, baseball, etc), writing utensil, writing pad, and access to the internet (if initial velocity is unknown)

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