Question
A bullet shot hits the ground 4km away, which was at an angle of 30° with the plane. What should be the projection angle to hit a target 6km away ? Neglect air resistance and assume it's muzzle speed
Answers
GPT 3.5
Let's call the projection angle we're trying to find θ.
In the first scenario, the bullet is shot 4km away at an angle of 30°. We can use the range formula to find the muzzle speed of the bullet:
range = (muzzle speed)^2 * sin(2θ) / g,
where g is the acceleration due to gravity.
In this case, the range is 4km (4000 m), and the angle θ is 30°. Assuming the acceleration due to gravity is 9.8 m/s², let's solve for the muzzle speed:
4000 = (muzzle speed)^2 * sin(2 * 30°) / 9.8
4000 * 9.8 = (muzzle speed)^2 * sin(60°)
39200 / sin(60°) = (muzzle speed)^2
(39200 / sin(60°)) = (muzzle speed)^2
muzzle speed = sqrt(39200 / sin(60°))
Now, in the second scenario, we want to hit a target 6km (6000 m) away. To find the projection angle θ, we can rearrange the range formula:
range = (muzzle speed)^2 * sin(2θ) / g
Solving for sin(2θ):
sin(2θ) = range * g / (muzzle speed)^2
Substituting the given values:
sin(2θ) = 6000 * 9.8 / sqrt(39200 / sin(60°))^2
sin(2θ) = 6000 * 9.8 / (39200 / sin(60°))
sin(2θ) = 6000 * 9.8 * sin(60°) / 39200
sin(2θ) = 0.866 / 6.28
sin(2θ) ≈ 0.1379
Next, we can find 2θ by taking the inverse sine of 0.1379:
2θ ≈ sin^(-1)(0.1379)
θ ≈ sin^(-1)(0.1379) / 2
Calculating the projection angle using a calculator, we find:
θ ≈ 7.92°
Therefore, the projection angle needed to hit the target 6km away is approximately 7.92°.
In the first scenario, the bullet is shot 4km away at an angle of 30°. We can use the range formula to find the muzzle speed of the bullet:
range = (muzzle speed)^2 * sin(2θ) / g,
where g is the acceleration due to gravity.
In this case, the range is 4km (4000 m), and the angle θ is 30°. Assuming the acceleration due to gravity is 9.8 m/s², let's solve for the muzzle speed:
4000 = (muzzle speed)^2 * sin(2 * 30°) / 9.8
4000 * 9.8 = (muzzle speed)^2 * sin(60°)
39200 / sin(60°) = (muzzle speed)^2
(39200 / sin(60°)) = (muzzle speed)^2
muzzle speed = sqrt(39200 / sin(60°))
Now, in the second scenario, we want to hit a target 6km (6000 m) away. To find the projection angle θ, we can rearrange the range formula:
range = (muzzle speed)^2 * sin(2θ) / g
Solving for sin(2θ):
sin(2θ) = range * g / (muzzle speed)^2
Substituting the given values:
sin(2θ) = 6000 * 9.8 / sqrt(39200 / sin(60°))^2
sin(2θ) = 6000 * 9.8 / (39200 / sin(60°))
sin(2θ) = 6000 * 9.8 * sin(60°) / 39200
sin(2θ) = 0.866 / 6.28
sin(2θ) ≈ 0.1379
Next, we can find 2θ by taking the inverse sine of 0.1379:
2θ ≈ sin^(-1)(0.1379)
θ ≈ sin^(-1)(0.1379) / 2
Calculating the projection angle using a calculator, we find:
θ ≈ 7.92°
Therefore, the projection angle needed to hit the target 6km away is approximately 7.92°.