A rifle that has been "sighted in" for a 94.6-meter target. If the muzzle speed of the bullet is v0 = 429 m/s, what are the two possible angles è1 and è2 between the rifle barrel and the horizontal such that the bullet will hit the target? One of these angles is so large that it is never used in target shooting
To find the two possible angles, we can use the kinematic equations of motion for projectile motion. The key equation we will use is:
y = x * tan(θ) - (g * x^2) / (2 * v0^2 * cos^2(θ))
where:
- y is the vertical distance from the target (94.6 meters)
- x is the horizontal distance to the target (94.6 meters)
- θ is the launch angle (in radians)
- g is the acceleration due to gravity (9.8 m/s^2)
- v0 is the muzzle speed of the bullet (429 m/s)
Substituting the given values into the equation, we can rearrange it to solve for θ:
94.6 = 94.6 * tan(θ) - (9.8 * 94.6^2) / (2 * 429^2 * cos^2(θ))
Simplifying further:
tan(θ) - (9.8 * 94.6) / (2 * 429^2 * cos^2(θ)) = 1
Now, we need to use numerical methods such as the numerical root-finding method, like Newton's method or bisection method, to solve for the two possible launch angles θ1 and θ2 that satisfy the equation. These numerical methods involve iterative calculations to find the values of θ that make the equation equal to 1.
Keep in mind that one of the two angles (θ2) will be larger than 90 degrees (π/2 radians) since it's mentioned that it is never used in target shooting.
Using a numerical method or numerical software such as MATLAB or Python with numerical libraries can help you find the values of θ1 and θ2.
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