Explain procedure to Determination of height of a tree/ building and horizontal/ vertical distance by sextant

 A sextant is a navigational instrument traditionally used to measure the angle between two visible objects. It can also be used to determine the height of a tree or building and to measure horizontal and vertical distances. Here’s a detailed procedure for using a sextant for these purposes:

Determining the Height of a Tree or Building

1. Find a Suitable Location:

   • Choose a point at a known distance ($d$) from the base of the tree or building. The distance should be measured horizontally on level ground.

2. Measure the Angle of Elevation:

   • Stand at the chosen point and sight the top of the tree or building through the sextant.
   • Adjust the index mirror of the sextant until the image of the top of the tree or building aligns with the horizon.
   • Read the angle of elevation ($\theta$) from the sextant’s scale.

3. Calculate the Height:

   • Use trigonometric principles to calculate the height ($h$) of the tree or building above your eye level.
   • The height can be calculated using the formula: $h = d \cdot \tan(\theta)$
   • If your eye level is at a height ($e$) above the ground, add this to the calculated height: $\text{Total height} = h + e$

Measuring Horizontal and Vertical Distances

Horizontal Distance

1. Initial Setup:

   • Choose two points, $A$ and $B$, on level ground with a clear line of sight between them.
   • Measure the horizontal distance ($d$) between points $A$ and $B$.

2. Sight the Target:

   • From point $A$, sight the top of the object (e.g., a tree or building) using the sextant.
   • Adjust the index mirror until the image aligns with the horizon and read the angle of elevation ($\theta_A$).

3. Move to the Second Point:

   • Move to point $B$ and repeat the sighting process to obtain the angle of elevation ($\theta_B$).

4. Calculate the Horizontal Distance:

   • Using the angles obtained, apply trigonometric calculations to determine the horizontal distance ($D$) from each point to the base of the object. For simplicity, we consider: $D = \frac{d}{\tan(\theta_B) - \tan(\theta_A)}$

Vertical Distance

1. Initial Setup:

   • Stand at a point with a clear line of sight to both the base and top of the object.

2. Measure the Angles:

   • Measure the angle of elevation to the top of the object ($\theta_{top}$).
   • Measure the angle of depression to the base of the object ($\theta_{base}$).

3. Calculate the Vertical Distance:

   • The vertical distance ($H$) can be calculated by first determining the horizontal distance ($D$) to the base using the angle of depression.
   • Using the angle of elevation: $H = D \cdot (\tan(\theta_{top}) + \tan(\theta_{base}))$

Practical Considerations

• Accuracy of Measurements: Ensure the sextant is properly calibrated and that you measure angles accurately.
• Level Ground: Ensure the ground between measurement points is level to avoid errors in distance calculation.
• Clear Line of Sight: Make sure there are no obstructions between your observation points and the object being measured.

• Determining the Horizontal Distance Using a Sextant

  Procedure

  1. Position Yourself: Choose two points along a straight line from the object at known distances $D_1$ and $D_2$ apart.

  2. Measure Angles: Use the sextant to measure the angles of elevation ($\theta_1$ and $\theta_2$) from these points to the top of the object.

  3. Calculate the Distance:

     • Using the angles and the known distance between the two points ($D_1$ and $D_2$), apply the formula for determining the horizontal distance ($D$): $$D = \frac{D_2 \cdot \tan(\theta_1) - D_1 \cdot \tan(\theta_2)}{\tan(\theta_1) - \tan(\theta_2)}$$

  Determining the Vertical Distance Using a Sextant

  Procedure

  1. Position Yourself: Stand at a point where you can measure the angle of depression or elevation.

  2. Measure the Angle: Use the sextant to measure the angle of depression ($\alpha$) or elevation ($\theta$) to the point directly above or below you.

  3. Calculate the Vertical Distance:

     • For the angle of depression, the vertical distance ($V$) can be calculated using: $$V = D \cdot \tan(\alpha)$$
     • For the angle of elevation, the vertical distance ($V$) is: $$V = D \cdot \tan(\theta)$$

  In these calculations, $D$ is the horizontal distance from the observer to the point directly below or above the object.

  By carefully measuring the angles and distances and applying these trigonometric principles, a sextant can be an effective tool for determining the height of objects and horizontal or vertical distances.