Why does the height you drop a ball from onto sand, affect the width of the crater?

It's down to kinetic energy.

KE = 1/2 * m * v^2 (where v is the speed of your falling ball).

The width of the crater gets bigger as the speed of the falling ball increases because more kinetic energy is imparted to (and hence absorbed by) the sand.

I'm not sure if this answer is over-simplified, but when you are moving a ball up (e.g. lifting it up to a height), this gives the ball more potential energy. The higher the ball goes, the more potential energy that the ball receives. Once the ball is dropped, the potential energy turns into kinetic (movement) energy. Once it reaches the ground, all the potential energy has turned to movement. The ball has reached the ground, so there is still movement energy in the ball, but it can't go any further. Because of this, the energy is transferred into the sand, and the sand grains have energy to move away from the ball.

Going back to the main question - the higher the ball is, the more energy is put into it. Once the ball reaches the ground, there is more energy, so the sand grains have more energy to move, and can move further away than with a lower height.

Every body or object has a terminal velocity, and that means every object has a maximum speed it cannot go past unless you give it extra energy. This only applies when the object is in free fall, which means if the object is left to gravity.

For example if you drop a ball from the top of a roof, only gravity and air resistance are acting on it. At first only gravity is acting on the ball, so the ball is travelling at 9.81 metres per second per second. 9.81 Newtons per kilogram is a constant number that applies to all objects on the Earth. Air resistance then starts acting on the ball, trying to slow it down. After a while the forces of air resistance and gravity balance so that the ball is travelling as fast as it can without you giving it more energy.

Now to answer your question:

When you raise an object from the ground, it gains gravitational potential energy. As i explained earlier each object has its own terminal velocity. But for the forces of air resistance (also called drag) and gravity to balance takes time. So that means every object has a certain and specific height it has to be dropped from for it to have enough time for the two forces to balance. When you drop the object, the gravitational potential energy is changed into kinetic energy.

It is also useful to know that sometimes there is a relationship between gravitational potential energy (GPE) and kinetic energy (KE). The gain in KE is equal to the loss of GPE. This means that when GPE is at its highest, there is no KE. And when KE is at its highest, there is no GPE. When there is no GPE, the object is on the ground.

As the object, the ball, drops, it gains more and more KE. So when it hits the ground, it has the most KE it can gain, provided it gained its terminal velocity. When the ball hits the ground, this KE is changed into heat energy and sound energy, among other things. The impact has a lot of energy and therefore the crater the ball makes will be proportionally big.

If a ball has gained its terminal velocity, the crater with be at its largest. If the ball is travelling at a slower velocity, the crater won't be as big.

The force the ball hits the ground can be calculated using the equation F=ma (Force=mass x acceleration) where F is the Force of the ball, m is the mass of the ball and a is the acceleration the ball had acting on it, which would be 9.81 if the ball was in free fall.

Kinetic energy can be calculated using KE=0.5 x m x v^2 (Kinetic energy= 0.5 x the mass of the ball x the square of the velocity it was travelling at). ONLY THE VELOCITY IS SQUARED.

Gravitational Potential Energy can be calculated using GPE=m x g x h (Gravitational Potential Energy=the mass of the object x the gravitational constant (which is always 9.81) x the how high up the ball was from the ground).

The units of height are always in metres (m).

The units of mass are always in kilograms (kg).

The units of acceleration are always in metres per second squared (ms^-2).

The units of force are always in Newtons (N).

The units of the gravitational constant, although i don't think you'll be asked this, are in newtons per kilogram (NK^-1).

The units of energy are always in joules (J).

If any of the units in a question aren't as i've said in a question you'd have to convert them into the correct ones for the equations to work properly.

Only up to a point. Once it has achieved it's terminal velocity, dropping from a greater height will not make any change to the energy of impact.

for every action there is a reaction . try dropping a hammer on your foot and youl get the meaning.