A laser is mounted on a table 1.04m above the ground, pointed at a vertical mirror a horizontal distance 3.46m away. The laser beam strikes the mirror 1.52m above the ground. Behind the laser is a wall, 2.80m from the aperture where the laser light emerges.
At what height does the reflected beam strike the wall?
To find the height at which the reflected beam strikes the wall, we can use the concept of similar triangles.
Let's label the points involved in the problem:
- Aperture: A
- Mirror: M
- Wall: W
We know the following measurements:
- Distance from the aperture to the mirror: AM = 3.46 m
- Height of the aperture above the ground: AH = 1.04 m
- Height at which the laser beam strikes the mirror: MH = 1.52 m
- Distance from the aperture to the wall: AW = 2.80 m
Since the laser beam is reflected off the mirror and strikes the wall, we can form similar triangles AMH and WMX (where X is the point where the reflected beam strikes the wall).
Using the concept of similar triangles, we can set up the following proportion:
AM / MH = WM / XH
Substituting the known values:
3.46 m / 1.52 m = (distance from the mirror to the wall) / (height at which the reflected beam strikes the wall)
Let's solve the proportion for XH, the height at which the reflected beam strikes the wall:
(3.46 m / 1.52 m) = (distance from the mirror to the wall) / XH
Cross-multiplying:
3.46 m * XH = 1.52 m * (distance from the mirror to the wall)
Simplifying:
XH = (1.52 m * distance from the mirror to the wall) / 3.46 m
Substituting the given value for the distance from the mirror to the wall (AW = 2.80 m):
XH = (1.52 m * 2.80 m) / 3.46 m
Calculating:
XH = 1.2248 m
Therefore, the reflected beam strikes the wall at a height of approximately 1.2248 meters.
To determine at what height the reflected beam strikes the wall, we can use the concept of the Law of Reflection. According to the Law of Reflection, the angle of incidence (i) is equal to the angle of reflection (r), and both angles are measured with respect to the normal line (a line perpendicular to the surface of the mirror).
Let's break down the given information to solve this problem step by step:
1. The laser is mounted on a table 1.04m above the ground. Therefore, the height of the laser beam when it strikes the mirror is 1.04m.
2. The vertical mirror is a horizontal distance of 3.46m away from the laser. This means that the horizontal distance between the laser's aperture and the point where the reflected beam strikes the wall is also 3.46m.
3. The laser beam strikes the mirror 1.52m above the ground. This gives us the height of the point of incidence on the mirror.
4. Behind the laser is a wall, located 2.80m from the aperture where the laser light emerges.
Now, let's calculate the height at which the reflected beam strikes the wall:
Step 1: Calculate the angle of incidence (i).
- The angle of incidence can be determined using the height of the laser beam above the ground and the horizontal distance between the laser and the mirror. We can use trigonometry to find this angle.
- Using the given information, the angle of incidence (i) is given by:
i = tan^(-1)(height of the laser beam / horizontal distance)
i = tan^(-1)(1.04m / 3.46m)
Step 2: Calculate the angle of reflection (r).
- Since the angle of incidence is equal to the angle of reflection, we can use the value of the angle of incidence (i) calculated in Step 1.
Step 3: Calculate the height at which the reflected beam strikes the wall.
- We can use the angle of reflection (r) and the horizontal distance between the laser and the mirror to find the height at which the reflected beam strikes the wall.
- Using trigonometry, the height at which the reflected beam strikes the wall is given by:
height on the wall = horizontal distance * tan(r)
By following these steps and plugging in the values, you can calculate the height at which the reflected beam strikes the wall.