A rocket shoots straight up from the launch pad.Five seconds after lift off, an observer 2 miles away notes that the rocket's angle of elevation is 41 degrees.How far did the rocket rise during those 4 seconds?
To determine the distance that the rocket rose during those 4 seconds, we can use trigonometry.
First, let's draw a diagram to visualize the situation. We have a right triangle formed by the observer, the rocket, and the ground. The rocket's initial height is unknown, but we can assume it is negligible compared to its final height.
Let's denote the vertical distance the rocket rose during those 4 seconds as 'h'. The distance between the observer and the launch pad is given as 2 miles. We can use the tangent function to relate the angle of elevation (41 degrees) to the ratio of the vertical distance (h) to the horizontal distance (2 miles). The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
Therefore, we have the trigonometric equation:
tan(41 degrees) = h / 2 miles
To find the value of h, we can rearrange the equation as follows:
h = 2 miles * tan(41 degrees)
Now, we can calculate the value of h using a scientific calculator:
h ≈ 2 miles * tan(41 degrees)
h ≈ 2 miles * 0.869
h ≈ 1.738 miles
So, the rocket rose approximately 1.738 miles during those 4 seconds.
To find the distance the rocket has risen during those 4 seconds, we need to break down the problem into steps:
Step 1: Determine the initial height of the rocket.
Since the observer notes the rocket's angle of elevation 5 seconds after lift-off, we know that the rocket has already risen a certain height. To find this initial height, we can use the trigonometric relationship between angle of elevation and height.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the initial height of the rocket, and the adjacent side is the distance from the observer to the rocket.
Using trigonometry, we have:
tan(41 degrees) = (initial height) / 2 miles
We can solve this equation to find the initial height:
(initial height) = tan(41 degrees) * 2 miles
Step 2: Determine the rocket's final height after 4 seconds.
During the 4 seconds in question, the rocket continues to rise. We need to find the increase in height during this time.
Given that the rocket's speed is constant and it shoots straight up, we can use the equation of motion for uniformly accelerated vertical motion:
final height = initial height + (initial velocity × time) + (0.5 × acceleration × time^2)
Since the rocket shoots straight up, the initial velocity is 0 (assuming the initial velocity is directed horizontally). Additionally, the acceleration due to gravity is -32 ft/s^2 (negative because it is acting downward).
Converting the time of 4 seconds to seconds:
4 seconds = 4 * 5280 ft / 1 mile * 1 hour / 3600 sec = 352 / 3600 ft = 88 / 900 ft
Final height = (initial height) + (0 × 88 / 900 ft) + (0.5 × -32 ft/s^2 × (88 / 900 ft))^2
Step 3: Convert the final height to miles.
To find the distance the rocket has risen during those 4 seconds, we need to convert the final height from feet to miles.
final height (miles) = final height (feet) / 5280 ft / 1 mile
Now, let's calculate the values:
(initial height) = tan(41 degrees) * 2 miles
(final height) = (initial height) + (0 × 88 / 900 ft) + (0.5 × -32 ft/s^2 × (88 / 900 ft))^2
(final height in miles) = (final height in feet) / 5280 ft / 1 mile
Note: To obtain accurate results, the trigonometric calculations should be done using radians rather than degrees. However, since the values are provided in degrees, we will use degrees for simplicity in this calculation. The resulting value will be approximate.
Now, let's calculate the values:
(initial height) = tan(41 degrees) * 2 miles
(final height) = (initial height) + (0 × 88 / 900 ft) + (0.5 × -32 ft/s^2 × (88 / 900 ft))^2
(final height in miles) = (final height in feet) / 5280 ft / 1 mile
Plug in the values:
(initial height) ≈ tan(41 degrees) * 2 miles ≈ 1.440514 * 2 miles ≈ 2.881028 miles
(final height) ≈ (2.881028 miles) + (0 × 88 / 900 ft) + (0.5 × -32 ft/s^2 × (88 / 900 ft))^2 ≈ 2.881028 miles
(final height in miles) ≈ (2.881028 miles) / 5280 ft / 1 mile ≈ 2.881028 / 5280 miles ≈ 0.000546 miles
Therefore, the rocket rises approximately 0.000546 miles during those 4 seconds.