Enrol to start learning
Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.
1.6. Example 1.24: Horizontal Distance and RL Calculation with Tacheometer
Learn content
Interactive Audio Lesson
Unlock the classroom podcast
The transcript is free to read. A free account plays the conversation back.
Today, we're going to explore tacheometry! It's an efficient way to determine distances and levels. Can anyone recall what a tacheometer does?
Isn’t it a device that measures angles and distances at the same time?
Exactly! It uses angular measurements to calculate horizontal distances. Tacheometers can significantly speed up the process of measuring large distances. Let's introduce some key terms: K is the constant used for distance calculations, and C is often a constant for vertical measurements.
So, what do these constants actually mean in practice?
Great question! K helps convert the stadia readings to distances, while C accounts for any offsets in the height of the instrument. Remember: K is always multiplied with the difference in readings to get the distance!
How do we actually calculate the RL with these readings?
We collect the elevation data from the staff readings and consider the height of the instrument to find the RL. Let's recap: Understanding K and C is vital for our calculations.
Unlock the classroom podcast
The transcript is free to read. A free account plays the conversation back.
Now let’s dive into the calculations for vertical and horizontal distances. We will work through an example using the staff readings from our observations.
What are the first steps in making those calculations?
First, we calculate the staff differences by subtracting the lower reading from the upper reading. For instance, if our readings are 2.600 m and 1.200 m, the difference is crucial.
And then we use those differences in formulas for vertical distance?
Correct! We apply the formulas that incorporate the angle of depression or elevation. Remember: Vertical distance = KS sin²(θ)/2 + C sin(θ).
And after calculating the vertical distance, how do we get the horizontal distance?
For horizontal distance, we use D = KS cos²(θ) + C sin(θ). Always align your angles based on whether they indicate elevation or depression!
Unlock the classroom podcast
The transcript is free to read. A free account plays the conversation back.
Now that we have our distances, let's find the RL of the observed point B. Could someone summarize how we combine our data?
We'll add the height of the instrument and subtract the vertical distance observed, right?
Exactly! The formula is: RL of B = RL of the Bench Mark + height of instrument - vertical distance. Why is it critical to keep track of these values?
Because it ensures accuracy in elevation calculation!
Correct! A mistake in any measurement can lead to significant errors in RL calculations. Always double-check your tacheometric constants and observation data!
Unlock the classroom podcast
The transcript is free to read. A free account plays the conversation back.
Finally, let’s talk about where we can apply tacheometry in real-life scenarios. Can you think of any?
Maybe in construction for determining site levels?
Absolutely! It’s used for site surveys, road construction, and even in mining. Its speed and accuracy make it invaluable.
What about in layout and planning?
Exactly! Engineers rely on tacheometry for both layout and long distances in topographic surveys. It's a key tool in modern surveying.
Overview
Short Summary
This section discusses the use of a tacheometer for calculating horizontal distance and reduced level (RL) using angular measurements from a vertical staff.
Medium Summary
It explains the procedure for computing horizontal distances and reduced levels (RL) from tacheometer readings, demonstrating how the constants are utilized in the calculations.
Detailed Summary
Detailed Summary
In this section, the focus is on using a tacheometer for horizontal distance and reduced level (RL) calculations. A tacheometer is a surveying instrument that facilitates fast distance measurement by utilizing angular measures. The example provided involves observations taken at a staff station and specifics on how these readings, along with the constants of the instrument, are applied to deduce both the horizontal distance to a point and the RL of that point. The procedure includes:
- Defining Key Variables: Understanding the constants of the instrument, particularly the constant K and C, which are critical in calculations.
- K = 100
- C = 0.15
- Calculating Vertical Distances: Using tacheometric formulas which consider the angles of depression and elevation for two observations to derive vertical distances.
- Calculating Horizontal Distance: The angle of sight is manipulated to compute the corresponding horizontal distances.
- Determining RL: The RL is then calculated using the instrument height and the obtained vertical distance, culminating in the final reduced level of the point being observed.
This section emphasizes practical applications of tacheometry in surveying, highlighting the efficiency of the method in spatial data collection.
Audio Book
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountA tachometer was setup at a station A and the following readings were obtained on a staff held vertical. Calculate the horizontal distance AB and RL of B, when the constants of instrument are 100 and 0.15.
Detailed Explanation
In this example, a tachometer is set up at a point designated as station A to measure the distance and elevation related to a point called B. The readings obtained from a staff (a vertical measuring stick) are crucial. The constants used in the calculations are also provided — K Equals 100 and C Equals 0.15.
Examples & Analogies
Imagine you're standing at a height (like on a lookout tower) and you're trying to measure the distance to a tree while marking the height difference. The tachometer acts like your eyes giving you the readings of the tree's height and distance based on the angles observed.
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountIn first observation S = 2.600 – 1.200 = 1.400 m \theta = - 60 40’ (Depression) K = 100 and C = 0.15 Vertical Distance V = KS sin²θ / 2 + C sinθ = 100 (1.400) sin (2 x 60 40’) / 2 + 0.15 sin 60 40’ = 16.143 + 0.0174 = 16.160 m
Detailed Explanation
In the first observation, we calculate the vertical distance (V) based on the readings. The value of S is determined by subtracting the staff reading at the bottom from that at the top. The angle θ indicates that the line of sight is downward (depression). We plug in these values into the formula for vertical distance which combines K and C constants along with the sine of the angle to compute V.
Examples & Analogies
Think of this as measuring how far you are looking down from a balcony to a garden below. The higher you are, the steeper the angle, and thus the distance you calculate gets affected by how far down you are viewing things.
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountIn second observation S = 2.400 – 0.800 = 1.600 m \theta = + 80 20’ (Elevation) Vertical Distance V = KS sin²θ/2 + C sinθ = 100 (1.600) sin (2 x 80 20’) / 2 + 0.15 sin 80 20’ = 22.944 + 0.022 = 22.966 m
Detailed Explanation
For this observation, we again calculate the vertical distance looking upwards, which has a positive angle (elevation). Here, the process involves putting together the new staff readings and the upward angle similarly to how we did in the first observation but now considering that we're looking up.
Examples & Analogies
Visualize looking up at a tall building from the street. The elevation ensures you calculate a different height because you're adding the distance you see upward from the ground to the height of your eye level.
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountHorizontal distance D = KS cos²θ + C sinθ = 100 (1.600) cos² 80 20’ + 0.15 sin 80 20’ = 156.639 + 0.148 = 156.787 m
Detailed Explanation
Here, the final horizontal distance D is calculated by combining the horizontal components derived from the angle and measuring constants. It transforms the vertical measurements into a flat distance which is crucial for constructing accurate plans.
Examples & Analogies
Imagine you are kicking a soccer ball. The distance it travels on the ground (horizontal distance) will vary depending on how high you kick it (the angle) and how strong you kick it (the constants). The higher you aim, the less distance it may cover. This represents the relationship we see in our calculation.
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountRL of instrument axis = RL of BM + h + V = 850.500 + 1.900 + 16.160 = 868.560 m RL of B = RL of Inst. axis + V – h = 868.560 + 22.966 – 1.600 RL of B = 889.926m
Detailed Explanation
To find the final Reduced Level (RL) of point B, we begin with the RL of the Benchmark (BM) and add the vertical height and the previously calculated vertical distance V. Similarly, we calculate the RL of point B by using the instrument axis RL, added to the vertical distance and subtracting the height of the instrument.
Examples & Analogies
Consider it like starting from a known height (like the base of a mountain) and moving up to find another height at a different point. You begin with what you know (the base), add what you’ve climbed (vertical distance), and adjust based on where you are standing (height of the instrument). This summation gives you the height of the new location.
--
Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Tacheometry:
A method for surveying that relies on angular measurements to ascertain distances and elevations.
- Constants K and C:
Essential values used in the calculations for determining distances and vertical angles.
- Stadia Readings:
Measurements taken from a staff that aid in calculating distances rapidly.
Examples
Memory aids
Tacheometer, tall and sleek, measures angles with a peak; K for distance, C for height, getting RL is our delight!
Imagine a surveyor at a tall mountain. With his tacheometer, he sees down to a lake. He uses his readings and the instrument's constants, K and C, to find where the lake is in relation to his height!
K for Keep distance, C for Compensate height, V for Vertical calculations – remember KCV for tacheometer!
Flash Cards
Glossary
Tacheometer
A surveying instrument that measures distances and angles using stadia readings and surveys with a vertical staff.
Horizontal Distance
The distance measured along the horizontal plane from the tacheometer to the point of observation.
Reduced Level (RL)
The height of a point relative to a chosen datum, typically sea level.
Stadia Reading
The readings taken from a vertical staff held at a measured distance which helps in computing distances and heights in tacheometry.
Constants K and C
Values used in tacheometric calculations where K is a constant for distance calculations and C accounts for offsets.