Keeping an egg safe when you drop it is a project found in science classes around the country. The concept is that you have to change the properties of the egg to the point where it won't crack when dropped. This isn't as complicated as it sounds, and can be accomplished using the basic principles of science as well as a few things you probably have lying around your house.
Insert your egg into a jar, cup or some other type of container. Carefully place the egg into the container, as you don't want to break it before you've completed this process.
Fill your jar, cup or other container with regular vinegar to the point where it is completely covering the egg. Place this container in the refrigerator and leave it there for at least 24 hours.
Remove your container from the refrigerator. Dump out the vinegar and wash off the egg under your faucet. The vinegar will have disintegrated the shell on the egg, and it can now be dropped without being broken.
Place your raw egg (gently) inside a can large enough to fit the entire egg. Fill the can with salt water and cover it tightly. Cover the can (plastic tin can coverings can be purchased at stores, and you can use any watertight material). The egg will float in the salt water and, when dropped, the salt water will absorb the shock of the impact and protect the egg.
Place your egg inside a foam cup. Side another foam cup over top of the one that the egg is inside, allowing it to act like a cover. Put tape or another type of adhesive material on the two cups, securing them together. When dropped, the foam material used to make the cups will absorb the shock of the fall and prevent the egg from cracking.
Insert your egg into a regular balloon. Stretch out the mouth of the balloon to allow the egg to fit comfortably inside. Fill your balloon with water (enough to cover the egg) and tie it securely. When dropped, the water will absorb the impact of the fall and protect the egg from cracking or breaking.