### Video Transcript

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 ๐ equals 9.8 meters per second squared.

We can record the mass value of our stone, 520 grams, as ๐. And weโll write the distance the stone has fallen from rest, 8.1 meters, as ๐. Weโll also record the acceleration due to gravity, 9.8 meters per second squared. We want to solve for the stoneโs momentum at this point, after it has fallen 8.1 meters down. Weโll call this momentum capital ๐ป.

To begin on our solution, letโs recall the mathematical relationship describing ๐ป. The momentum of a massive body ๐ป is equal to the mass of that object multiplied by its speed. In our scenario, weโre given the mass of our object, the falling stone. And we want to solve for its speed, ๐ฃ. We know that the following stone is under the influence of a constant acceleration. The acceleration due to gravity, ๐. 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, ๐, 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 weโve been given in our problem statement. If we call ๐ฃ sub ๐ the initial velocity of the stone and ๐ฃ its velocity after itโs fallen 8.1 meters, then we can say that ๐ฃ sub ๐ is equal to zero because we assume the stone is released from rest. So our equation becomes: ๐ฃ squared is equal to two times ๐ times ๐. Or, ๐ฃ equals the square root of two ๐๐.

Plugging this expression in for ๐ฃ in our equation for momentum ๐ป, we can see weโve been given the mass ๐, the distance ๐, and the acceleration due to gravity ๐. So weโre ready to plug in and solve for ๐ป. When we do, weโre 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 ๐ป is equal to 6.552 kilograms meters per second. Thatโs the momentum of our mass after itโs fallen 8.1 meters.