a rifle that has been ‘sighted in' for a 91.4-meter target. If the muzzle speed of the bullet is v0 = 328 m/s, there 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. Give your answers as (a) the smaller angle and (b) the larger angle.
From d = Vo^2[sin(2µ)]/g
91.4 = 328^2[sin(2µ)]/9.8, µ = .2385º or 89.7615º.
do you just subtract from 90 to get the other angle
To determine the two possible angles (è1 and è2) between the rifle barrel and the horizontal, we can use basic principles of projectile motion and trigonometry. Here's how you can calculate these angles step by step:
Step 1: Establish the given information:
- Muzzle speed of the bullet (v0) = 328 m/s
- Distance to the target (d) = 91.4 meters
Step 2: Split the problem into horizontal and vertical components:
Since we are looking for angles, we separate the initial velocity (v0) into its horizontal and vertical components.
- Horizontal component (v0x) remains constant throughout the motion and can be found using the formula v0x = v0 * cos(θ).
- Vertical component (v0y) is affected by the acceleration due to gravity and can be found using the formula v0y = v0 * sin(θ), where θ is the angle we need to determine.
Step 3: Calculate the time of flight (t):
The time it takes for the bullet to reach the target can be found using the formula t = d / v0x, as the horizontal velocity remains constant.
Step 4: Determine the vertical displacement (y):
To find the vertical displacement, we can use the formula y = v0y * t + (1/2) * g * t^2, where g is the acceleration due to gravity (~9.8 m/s^2).
Step 5: Solve for the angles (è1 and è2):
- To find the smaller angle (è1), use the inverse trigonometric function tan(θ) = y / d.
- To find the larger angle (è2), subtract è1 from 90 degrees.
Finally, we can plug in the given values and solve the equations numerically to find the answers.
Note: I'll provide the numerical values for completeness, but it's always good to work with variables until the final step.
Step 2: Calculate the horizontal component:
v0x = 328 m/s * cos(θ)
Step 3: Calculate the time of flight:
t = 91.4 m / v0x
Step 4: Determine the vertical displacement:
y = (328 m/s * sin(θ)) * t - (1/2) * 9.8 m/s^2 * t^2
Step 5: Solve for the angles:
tan(θ) = y / 91.4 m
Find θ using the inverse tangent function.
Using these steps, you can determine both angles (è1 and è2) and then identify the smaller and larger angles as per the question.