Two rifles are fired at targets, which are 100 m away. The first rifle fires bullets at 1200 ft/s and the second fires a bullet at 3000 ft/s. How long will it take each bullet to get to the target? If both are aimed directly at the bull's-eye, how far will each bullet travel below the bull's-eye?

Question

Two rifles are fired at targets, which are 100 m away. The first rifle fires bullets at 1200 ft/s and the second fires a bullet at 3000 ft/s. How long will it take each bullet to get to the target? If both are aimed directly at the bull's-eye, how far will each bullet travel below the bull's-eye?

Answer 1

let α be the angle of the projectile with the horizontal as it leaves the muzzle.

Here α=0 degree.
Let vi=initial muzzle velocity, then
horizontal velocity, vh=vi*cos(α).

Time taken to reach the target
t=Horizontal distance/vh
where horizontal distance = 100m

Since acceleration due to gravity acts on the bullet during this time, then
vertical displacement
Dv=vi*sin(α)t-(1/2)gt²
where Dv is negative downwards.

Use consistent units in the above equations, for example metres and seconds.

Answer 2

To calculate the time it takes for each bullet to reach the target, we can use the formula:

Time = Distance / Speed

First, let's convert the speed of the bullets to meters per second (since the distance is given in meters):

1 ft/s is approximately 0.3048 m/s

So, the speed of the first bullet is 1200 ft/s * 0.3048 m/s = 365.76 m/s
The speed of the second bullet is 3000 ft/s * 0.3048 m/s = 914.4 m/s

Now, we can calculate the time it takes for each bullet to reach the target:

For the first bullet:
Time = 100 m / 365.76 m/s ≈ 0.273 seconds

For the second bullet:
Time = 100 m / 914.4 m/s ≈ 0.109 seconds

Now, let's calculate how far each bullet will travel below the bull's-eye. Since both bullets are aimed directly at the target, the vertical displacement will depend on the time of flight for each bullet.

We can use the formula:

Vertical displacement = 1/2 * acceleration * time^2

The acceleration due to gravity can be approximated as -9.8 m/s^2 (negative because it acts in the opposite direction to the bullet's motion).

For the first bullet:
Vertical displacement = 0.5 * (-9.8 m/s^2) * (0.273 s)^2 ≈ -0.376 m

For the second bullet:
Vertical displacement = 0.5 * (-9.8 m/s^2) * (0.109 s)^2 ≈ -0.054 m

Therefore, the first bullet will travel approximately 0.376 meters below the bull's-eye, and the second bullet will travel approximately 0.054 meters below the bull's-eye.