Make sure your stand will prevent the cylinder of the bottle from rolling. Epoxy or hot glue the mirror near the middle of the bottle, along the length of the cylinder. Allow time for the glue to set. Fill the bottle with water before using it.
Take the bottle with the mirror and its stand outside along with the screen, the watch, and a pencil. Never look directly at the sun! Locate the sun in the sky, and place the bottle on its stand so that the mirror directly faces the sun. Place the viewing screen several meters away from the mirror, facing the mirror. You may need a partner to hold the viewing screen in place. Rotate the cylinder so that an image of the sun appears on the screen. You will get the best results when the mirror is perpendicular to the sun's rays, and the viewing screen is parallel to the mirror. This is easier to achieve when the sun is closer to the horizon. (Try this at different times of day and different times of year.) Observe the image of the sun on the screen. What do you notice?
Trace the image of the sun on the viewing screen, and start timing. Stop timing when the image of the sun has moved completely out of the circle you have traced. Try this several times. Vary the size of the image and time it again. What do you notice?
The tiny mirror is reflecting light from the sun and producing an image of the sun on the screen. There are two similar triangles in this experiment. One is an isosceles triangle whose base is the diameter of the sun, and whose congruent sides are rays coming from each side of the sun's diameter toward your mirror. The base of the second triangle is the diameter of the sun's image on the screen. Its congruent sides are rays coming from the mirror. Since these are similar triangles, the angular diameter of the image on the screen is the same as the angular diameter of the sun in the sky.
The earth's spinning causes the image of the sun to appear to move across the screen. What is the angular velocity of the earth's spin? If the earth spins 360 degrees per 24 hours, then it spins 15 degrees per hour or 0.25 degrees per minute. The time it takes for the sun's image to move "one sun diameter" is about 2 minutes. For larger or smaller images, the time will be constant, as it is a measure of another constant, the spin of the earth. The angular diameter of the sun can be found using a proportion:
15 degrees/60 minutes = x degrees/2 minutes
During periods of sunspot activity, you can safely observe sunspots this way. Remembering that this is a reflected image, can you find the location of the sunspots on the sun based on what you see on its image?