A hot-air balloon of mass 1.5 metric tons was accelerating vertically downward at 106.2 centimeters per second squared. Given that the acceleration due to gravity is 9.8 meters per second squared, find the lift force generated by the hot air.
We will begin by sketching a diagram of the balloon and the forces acting on it. We are told that the mass of the balloon is 1.5 metric tons. And we know there are 1000 kilograms in one ton. This means that the balloon has a mass of 1500 kilograms. There will be a downward force acting on the balloon equal to its weight, and we know this is equal to mass multiplied by gravity. 1500 multiplied by 9.8 is equal to 14700. We have a downward force acting on the balloon of 14700 newtons. We are trying to calculate the lift force generated by the hot air which we will call 𝐹.
The final piece of information we are given is that the balloon is accelerating vertically downwards at 106.2 centimeters per second squared. As there are 100 centimeters in one meter, this is equal to 1.062 meters per second squared. We will now use Newton’s second law, which states that the sum of our forces is equal to the mass multiplied by the acceleration. Taking the downward direction to be positive, the sum of the forces acting on the balloon is equal to 14700 minus 𝐹. This is equal to the mass 1500 multiplied by the acceleration 1.062.
The right-hand side simplifies to 1593. We can then subtract this as well as adding 𝐹 to both sides of our equation. 𝐹 is therefore equal to 14700 minus 1593, which in turn is equal to 13107.
The lift force generated by the hot air is 13107 newtons.