Calculate the momentum of a stone of mass 520 grams after it has fallen 8.1 meters vertically downwards. Consider the acceleration due to gravity to be 𝑔 equals 9.8 meters per second squared.
We can record the mass value of our stone, 520 grams, as 𝑚. And we’ll write the distance the stone has fallen from rest, 8.1 meters, as 𝑑. We’ll also record the acceleration due to gravity, 9.8 meters per second squared. We want to solve for the stone’s momentum at this point, after it has fallen 8.1 meters down. We’ll call this momentum capital 𝐻.
To begin on our solution, let’s recall the mathematical relationship describing 𝐻. The momentum of a massive body 𝐻 is equal to the mass of that object multiplied by its speed. In our scenario, we’re given the mass of our object, the falling stone. And we want to solve for its speed, 𝑣. We know that the following stone is under the influence of a constant acceleration. The acceleration due to gravity, 𝑔. This means that equations of motion known as the kinematic equations apply to the motion of our stone. Each one of the four kinematic equations assumes that acceleration, 𝑎, is constant throughout.
As we look through this list, we see that the second equation listed lets us solve for velocity based on acceleration and distance travelled, both of which we’ve been given in our problem statement. If we call 𝑣 sub 𝑖 the initial velocity of the stone and 𝑣 its velocity after it’s fallen 8.1 meters, then we can say that 𝑣 sub 𝑖 is equal to zero because we assume the stone is released from rest. So our equation becomes: 𝑣 squared is equal to two times 𝑔 times 𝑑. Or, 𝑣 equals the square root of two 𝑔𝑑.
Plugging this expression in for 𝑣 in our equation for momentum 𝐻, we can see we’ve been given the mass 𝑚, the distance 𝑑, and the acceleration due to gravity 𝑔. So we’re ready to plug in and solve for 𝐻. When we do, we’re careful to convert our mass from units of grams to units of kilograms, so that those units are consistent with those of the other variables in this expression.
When we enter this expression on our calculator, we find 𝐻 is equal to 6.552 kilograms meters per second. That’s the momentum of our mass after it’s fallen 8.1 meters.