To solve this, let's consider an object placed at two different distances from a light source (the torch), projecting two different images on the wall. We aim to check if the image heights are proportional to their corresponding distances.

Key Measurements and Variables:

• $d_1$: Distance of the object from the torch when the image height is $h_1$.
• $h_1$: Height of the image when the object is placed at $d_1$.
• $d_2$: Distance of the object from the torch when the image height is $h_2$.
• $h_2$: Height of the image when the object is placed at $d_2$.

General Principle:

This situation relies on the concept of similar triangles. The light from the torch spreads out, creating an image on the wall, and the relationship between the object’s distance from the light source and the image’s height should be proportional.

Thus, if we assume the object behaves like a point source light, the height of the image is proportional to the distance from the source. This means:

$\frac{h_1}{h_2} = \frac{d_1}{d_2}$

If this proportion holds true, then the heights of the images are in proportion to their respective distances from the torch.

Steps to Solve:

1. Measure Distances:

   • Measure $d_1$ and $d_2$, the distances of the object from the light source.

2. Measure Heights:

   • Measure $h_1$ and $h_2$, the heights of the images formed on the wall.

3. Check Proportion:

   • Calculate the ratio $\frac{h_1}{h_2}$ and $\frac{d_1}{d_2}$.
   • If these two ratios are approximately equal, then the heights of the images are in proportion to the distances.

Reflection and Pattern:

By performing this experiment with different objects and varying distances, you will likely notice a consistent pattern where the heights of the images and the distances from the light source maintain proportionality. This is because the torch acts as a point source of light, and the rays spread out uniformly, forming similar triangles with the object and its image.

Example (with values):

Let's assume:

• $d_1 = 2 \, \text{m}, \, d_2 = 4 \, \text{m}$
• $h_1 = 0.5 \, \text{m}, \, h_2 = 1.0 \, \text{m}$

Then:

$\frac{h_1}{h_2} = \frac{0.5}{1.0} = 0.5$ $\frac{d_1}{d_2} = \frac{2}{4} = 0.5$

Since $\frac{h_1}{h_2} = \frac{d_1}{d_2}$, the heights of the images are proportional to the distances.

Do you need help with any particular measurements or finding ratios? Let me know!

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Five Related Questions:

1. What happens to the size of the image if the object is moved even farther from the light source?
2. How does the brightness of the image change as the object is moved further from the torch?
3. Would the shape of the object affect the proportionality between height and distance?
4. Can you predict the size of the image if the distance is halved again?
5. What role does the angle of the light source play in this type of projection?

Tip:

When dealing with light projections, always remember that objects and images are related through similar triangles, so proportions often hold true under basic conditions.