Calculating the force in a wide range of situations is crucial to physics. Most of the time, Newton’s second law (F = ma) is all you need, but this basic approach isn’t always the most direct way to tackle every problem. When you’re calculating force for a falling object, there are a few extra factors to consider, including how high the object is falling from and how quickly it comes to a stop. In practice, the simplest method for determining the falling object force is to use the conservation of energy as your starting point.
Background: The Conservation of Energy
The conservation of energy is a fundamental concept in physics. Energy isn’t created or destroyed, just transformed from one form into another. When you use the energy from your body (and ultimately the food you’ve eaten) to pick up a ball from the ground, you’re transferring that energy into gravitational potential energy; when you release it, that same energy becomes kinetic (moving) energy. When the ball strikes the ground, the energy is released as sound, and some may also cause the ball to bounce back up. This concept is crucial when you need to calculate falling object energy and force.
The Energy at the Impact Point
The conservation of energy makes it easy to work out how much kinetic energy an object has just before the point of impact. The energy has all come from the gravitational potential it has before falling, so the formula for gravitational potential energy gives you all the information you need. It is:
E = mgh
In the equation, m is the mass of the object, E is the energy, g is the acceleration due to gravity constant (9.81 m s−2 or 9.81 meters per second squared), and h is the height the object falls from. You can work this out easily for any object that falls as long as you know how big it is and how high it falls from.
The Work-Energy Principle
The work-energy principle is the last piece of the puzzle when you’re working out the falling object force. This principle states that:
Average impact force × Distance traveled = Change in kinetic energy
This problem needs the average impact force, so rearranging the equation gives:
Average impact force = Change in kinetic energy ÷ Distance traveled
The distance traveled is the only remaining piece of information, and this is simply how far the object travels before coming to a stop. If it penetrates into the ground, the average impact force is smaller. Sometimes this is called the “deformation slow down distance,” and you can use this when the object deforms and comes to a stop, even if it doesn’t penetrate into the ground.
Calling the distance traveled after impact d, and noting that the change in kinetic energy is the same as the gravitational potential energy, the complete formula can be expressed as:
Average impact force = mgh ÷ d
Completing the Calculation
The hardest part to work out when you calculate falling object forces is the distance traveled. You can estimate this to come up with an answer, but there are some situations where you can put together a firmer figure. If the object deforms when it makes impact – a piece of fruit that smashes as it hits the ground, for example – the length of the portion of the object that deforms can be used as distance.
A falling car is another example because the front crumples from the impact. Assuming that it crumples in 50 centimeters, which is 0.5 meters, the mass of the car is 2,000 kg, and it is dropped from a height of 10 meters, the following example shows how to complete the calculation. Remembering that the average impact force = mgh ÷ d, you put the example figures in place:
Average impact force = (2000 kg × 9.81 m s−2 × 10 m) ÷ 0.5 m = 392,400 N = 392.4 kN
Where N is the symbol for a Newtons (the unit of force) and kN means kilo-Newtons or thousands of Newtons.
Working out the impact force when the object bounces afterward is a lot more difficult. The force is equal to the rate of change of momentum, so to do this you need to know the momentum of the object before and after the bounce. By calculating the change in momentum between the fall and the bounce and dividing the result by the amount of time between these two points, you can get an estimate for the impact force.