Motion of a Falling Ball
Who would ever need or want to know how fast or how long it may take for a ball to reach the ground if thrown upwards from a window? I would as well as you should! You may thing that this would be very difficult to determine but in all actuality the problem is quite simply. With a few equations and a little intuition anyone can answer these questions.
First you must understand a little about gravity and motion of a ball if thrown straight up into the air. As I would expect everyone knows that gravity is what keeps you here on earth. Many people overlook gravity and simply think of it as the force that holds us on the ground. Well I got news for you, there is more to it than that. Gravity is a constant acceleration towards the center of earth. This is constant anywhere on earth and does not vary in magnitude. Therefore we say that gravity has uniform acceleration. And from that we can say that the ball will have uniformly accelerated rectilinear motion.
Uniformly accelerated rectilinear motion is described as follows:
From there you find the velocity of the ball by integrating this equation to get:
where vo is the initial velocity, a in the
acceleration on the ball, and t is
time in seconds.
To now find the position of the ball at any time (t) we simply substitute the lemma for velocity, which is dx/dt. From here you can integrate the velocity function with respect to (t) to find the position of the ball at a time (t).
From here you can integrate the velocity function with respect to (t) to find the position of the ball at a time (t).
The equations that we have just integrated will allow for anyone to calculate the velocity, acceleration, position, and time of an object at any point. By solving for one variable and substituting it into another equation you can solve for one using the other.
Now that we know everything there is know about uniformly accelerated rectilinear motion and we have the equations needed to solve for possible variables. Let’s try to work a problem involving this motion.
I have also put together a real life simulation on the problem presented to us. Working Model allowed me to put together a universe that incorporated the effects of gravity and use it to illustrate the exact same problem that I solved for analytically. Click on the link below to watch the simulation.