A rocket is fired at an angle of 18.5 ° with the ground from a point 1670 m directly behind an observer. Shortly later the rocket is observed at an angle of elevation of 33.5 ° directly in front of and above the observer. How far on a direct line has the rocket moved?
Not enough to go on.
Depending on the initial speed, the rocket could be almost anywhere, on its way up or down, at a variety of heights.
If the initial speed was v, then you have the parabola and line
y = -1.112/v^2 x^2 + 0.335x
y = 0.662(x-1670)
The value of v can change everything.
To find the distance the rocket has moved, we can use trigonometry and the given information. Let's break down the problem step by step:
Step 1: Draw a diagram
Start by visualizing the scenario. Draw a diagram representing the observer's position, the initial position of the rocket, and its final position based on the given information.
Step 2: Identify the known values
From the problem, we know the following:
- The angle of elevation from the observer's position to the rocket is 33.5°.
- The angle between the ground and the initial path of the rocket is 18.5°.
- The distance between the observer and the starting point of the rocket is 1670 meters.
Step 3: Find the vertical and horizontal displacements
Using trigonometry, we can find the vertical and horizontal displacements of the rocket.
a) Vertical displacement (h):
The vertical displacement is the distance the rocket has moved directly above the observer. We can find h using the tangent of the angle of elevation:
tan(33.5°) = h / x, where x is the horizontal displacement.
b) Horizontal displacement (x):
The horizontal displacement is the distance the rocket has moved along the ground. We can find x using the tangent of the angle between the ground and the rocket's initial path:
tan(18.5°) = h / (x + 1670)
Step 4: Solve the equations
Now, we have two equations and two unknowns (h and x) that we can solve simultaneously.
From equation (a):
tan(33.5°) = h / x
Rearranging equation (b):
h = (x + 1670) * tan(18.5°)
Substituting h in equation (a):
tan(33.5°) = [(x + 1670) * tan(18.5°)] / x
Simplify the equation:
tan(33.5°) = (x * tan(18.5°) + 1670 * tan(18.5°)) / x
Now, cross multiply and solve for x:
x * tan(33.5°) = x * tan(18.5°) + 1670 * tan(18.5°)
Rearranging the equation:
x * (tan(33.5°) - tan(18.5°)) = 1670 * tan(18.5°)
Solving for x:
x = (1670 * tan(18.5°)) / (tan(33.5°) - tan(18.5°))
Step 5: Calculate the distance
Now, substitute the values into the equation to find x:
x = (1670 * tan(18.5°)) / (tan(33.5°) - tan(18.5°))
Calculating the value of x:
x ≈ 530.84 meters
Therefore, the rocket has moved approximately 530.84 meters on a direct line.