Velocity and Acceleration

When an object is moving there are several things to understand about its motion. We need to know how fast the object is moving, where it is going, and if it is speeding up or slowing down. In order to perceive motion we have to have a frame of reference which we use to determine that an object is moving. If we are inside of a car, for instance, and we look down at the floor we have a hard time determining if we are moving or not. If we look out the window we can easily tell that movement is occurring by watching objects on the side of the road move by. In order to detect motion we have to have a reference point that provides us with a set point that will move away or closer to us over time. Using this point as a reference we can detect how fast we are moving over a period of time.

Speed can be thought of as how far an object moves over a period of time. A fast-moving object has a high rate of speed and covers a distance in a short amount of time where a slow-moving object covers the same distance in a longer amount of time. This description is vague and does not tell us a lot about the object.

Let's take a baseball for example. If a batter hits the ball and it is traveling 100 miles/hour, we know how fast it is moving, but we have no idea where it is going. Imagine listening to a sports announcer and hearing the phrase "it's a speeding fly ball". That is not real exciting to listen to because it does not tell you a whole lot of information. This is where the announcer could use velocity to describe the balls motion in a way that presents you with more information. If we give the ball speed and a direction, then we have described velocity.

Using our baseball announcer again as an example we can see how using velocity can provide more information to the fans by giving the ball a direction. If the announcer states, "it's a speeding fly ball to center field" they are using velocity to describe the baseball. We not only know how fast it is moving, but where it is headed. If we look at this in mathematical terms we can describe the two quantities using the following formulas: $LaTeX: \text{speed}=\frac{\Delta\text{distance}}{\Delta\text{time}}$

Speed is equal to change (∆) in distance over change (∆) in time.

$LaTeX: \text{velocity}=\frac{\Delta\text{distance}}{\Delta\text{time + direction}}$

Velocity is equal to change (∆) in distance over change (∆) in time and direction.

Let's look at how important velocity is in a sporting event. If a player is running the ball in a football game and he is traveling at 5 yards per second, he could be running for a touchdown or he could be running towards the opposing team's end zone. This description of velocity is very important. It can mean the difference between scoring 7 points for your team and allowing the opposing team to score 2 points.

Let's look at one more quantity. If an object is speeding up or slowing down in a specific direction it is said to be accelerating. Think of a sprinter at a track meet, they start from rest at the starting line and they typically increase their velocity all the way to the finish line. The runner is increasing their velocity over time so they are said to have positive acceleration. After reaching the finish line they are decreasing in velocity and are said to have negative acceleration.

We can also look at this quantity in mathematical terms and use the formula below to describe acceleration. $LaTeX: \text{Acceleration}=\frac{\Delta\text{velocity}}{\Delta\text{time}}$

Acceleration is equal to change (∆) in velocity over change (∆) in time.

Another way to write the acceleration equation is: $LaTeX: \text{Acceleration}=\frac{\text{velocity}_f - \text{velocity}_i}{\text{time}_f - \text{time}_i}$

Where velocity_f is equal to the final velocity and velocity_i is equal to the initial velocity. Similarly time_f is the final time and time_i is the initial time. So, in the sprinter example above, if the runner was running the 100 meter dash, his initial velocity would be zero and his initial time would be zero because he starts the race motionless at the starting line. His final velocity might be 10 meters per second toward the finish and his final time might be 10.2 seconds. To solve, you would plug these numbers into the equation and solve as shown below. $LaTeX: \text{Acceleration}=\frac{\text{10 meters / second toward the finish - 0 meters / second}}{\text{10.2 seconds - 0 seconds}}\\ \text{Acceleration}=\frac{\text{10 meters / second toward the finish}}{\text{10.2 seconds}}\\ \text{Acceleration = 0.98m / s}^2$

***You might have noticed that the unit for acceleration will have a unit of time squared. This is because when the units are divided, the answer is 0.98 meters per second PER second. Instead of having double denominators, the 2nd per second flips up into the first denominator, and the denominator becomes seconds squared. You will not be tested over this concept.***

Watch the following presentation to learn how to solve motion problems.

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