An observer 2 km from the launching pad observes a vertically ascending missile at an angle of elevation of 21 degrees. 5 seconds later the nagle has increased to 35 degrees. What was its average speed during this interval? If it keeps going vertically at the same average speed, what will its angle of elevation be 15 seconds after the first sighting?
height at beginning of observation:
tan 21° = h/2 ----> height = 2tan21 = .76773
height after 5 seconds:
tan 35° = h/2 ----> height = 1.40042
change in height = .63269
average vertical speed = .63269/5 = .12654 km/s
so in 15 seconds it will have risen 1.89061 km on top of the .76773 km , its original height
so new height = 2.66579
tanØ = 2.66278/2 = 1.3328955..
Ø = 53.12°
To find the average speed of the missile during the 5-second interval, we can use trigonometry and the concept of average speed.
Let's denote the distance covered by the missile during the 5-second interval as "d" and the average speed as "v". We can use the concept of average speed, which is defined as the total distance divided by the total time.
During the 5-second interval, the observer observes the missile's angle of elevation changing from 21 degrees to 35 degrees. This change in angle implies that the missile has traveled vertically upwards.
We can consider the triangle formed by the observer, the launching pad, and the missile. The distance "d" covered by the missile can be calculated using the tangent function:
tan(35°) = (2 km + d) / x, where "x" is the horizontal distance covered by the missile.
tan(21°) = (2 km) / x
Combining these two equations, we can solve for "x" and hence find "d".
Let's calculate the values:
Using the equation tan(35°) = (2 km + d) / x, we can rewrite it as:
x = (2 km + d) / tan(35°) ---(1)
Using the equation tan(21°) = (2 km) / x, we can rewrite it as:
x = (2 km) / tan(21°) ---(2)
Now equating equation (1) and equation (2), we can solve for "d".
(2 km + d) / tan(35°) = (2 km) / tan(21°)
Cross-multiplying and rearranging, we get:
tan(21°)(2 km + d) = tan(35°)(2 km)
(2 km + d) = (tan(35°) / tan(21°))(2 km)
d = (tan(35°) / tan(21°))(2 km) - 2 km
Substituting the values and calculating, we get:
d ≈ 1.87 km
Now that we have the distance traveled by the missile in the 5-second interval, we can calculate the average speed "v" using the formula:
v = d / t
Where "t" is the time interval, which is 5 seconds in this case.
v = 1.87 km / 5 s
v ≈ 0.374 km/s
Therefore, the average speed of the missile during this interval is approximately 0.374 km/s.
Now, let's determine the angle of elevation after 15 seconds. Since the missile maintains the same average speed, we can assume it continues ascending vertically.
Using the average speed "v" calculated earlier, we can find the vertical distance covered by the missile in 15 seconds:
Vertical Distance = v * t
Vertical Distance = 0.374 km/s * 15 s
Vertical Distance ≈ 5.61 km
The vertical distance covered in 15 seconds can be equated to the distance between the observer and the missile after 15 seconds. Using this information, we can calculate the new angle of elevation:
tan(angle) = Vertical Distance / (2 km + Vertical Distance)
tan(angle) = 5.61 km / (2 km + 5.61 km)
tan(angle) ≈ 0.737
angle ≈ arctan(0.737)
angle ≈ 36.97 degrees
Therefore, the angle of elevation 15 seconds after the first sighting is approximately 36.97 degrees.
To find the average speed of the missile during the given interval, we need to calculate the change in vertical distance and the change in time.
First, let's find the change in vertical distance:
- At the initial observation, the angle of elevation is 21 degrees, and the observer is 2 km away from the launching pad. We can use trigonometry to find the initial vertical distance traveled by the missile.
The vertical distance traveled (d1) can be calculated as: d1 = 2 km * tan(21 degrees)
- After 5 seconds, the angle of elevation increases to 35 degrees. We can again use trigonometry to find the new vertical distance traveled.
The vertical distance traveled (d2) can be calculated as: d2 = d1 + (V * 5) * tan(35 degrees), where V is the average speed of the missile in km/s.
To find the average speed (V), we need to calculate the change in vertical distance over the change in time:
- Change in vertical distance: Δd = d2 - d1
- Change in time: Δt = 5 seconds
Average speed (V) = Δd / Δt
Now, let's calculate the average speed:
1. Calculate the initial vertical distance traveled (d1):
d1 = 2 km * tan(21 degrees)
2. Calculate the new vertical distance traveled (d2):
d2 = d1 + (V * 5) * tan(35 degrees)
3. Calculate the change in vertical distance (Δd):
Δd = d2 - d1
4. Calculate the average speed (V):
V = Δd / Δt
Once we find the average speed during the given interval, we can calculate the angle of elevation after 15 seconds.
To find the angle of elevation after 15 seconds:
1. Calculate the vertical distance traveled after 15 seconds (d3):
d3 = d2 + (V * 10) * tan(35 degrees), considering the additional time of 10 seconds (total 15 seconds).
2. Calculate the angle of elevation after 15 seconds (θ):
θ = arctan(d3 / (2 km))
By following these steps, we can determine the average speed during the given interval and the angle of elevation 15 seconds after the first sighting.