Question
To describe the motion of a particle along a straight line, it is often convenient to draw a graph representing the position of the particle at different times. This type of graph is usually referred to as an x vs. t graph. To draw such a graph, choose an axis system in which time t is plotted on the horizontal axis and position x on the vertical axis. Then, indicate the values of x at various times t Mathematically, this corresponds to plotting the variable x as a function of t An example of a graph of position as a function of time for a particle traveling along a straight line is shown below. Note that an x vs. t graph like this does not represent the path of the particle in space. What is the instantaneous velocity v of the particle at t = 10.0 s? Express your answer in meters per second. Another common graphical representation of motion along a straight line is the v vs. t graph, that is. the graph of (instantaneous) velocity as a function of time. In this graph, time t is plotted on the horizontal axis and velocity v on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only one nonzero component in the direction of motion. Thus, in this problem, we will call v the velocity and a the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion. Which of the graphs shown is the correct v vs. t plot for the motion described in the previous parts? To describe the motion of a particle along a straight line, it is often convenient to draw a graph representing the position of the particle at different times. This type of graph is usually referred to as an x vs. t graph. To draw such a graph, choose an axis system in which time t is plotted on the horizontal axis and position x on the vertical axis. Then, indicate the values of x at various times t Mathematically, this corresponds to plotting the variable x as a function of t An example of a graph of position as a function of time for a particle traveling along a straight line is shown below. Note that an x vs. t graph like this does not represent the path of the particle in space. What is the instantaneous velocity v of the particle at t = 10.0 s? Express your answer in meters per second. Another common graphical representation of motion along a straight line is the v vs. t graph, that is. the graph of (instantaneous) velocity as a function of time. In this graph, time t is plotted on the horizontal axis and velocity v on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only one nonzero component in the direction of motion. Thus, in this problem, we will call v the velocity and a the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion. Which of the graphs shown is the correct v vs. t plot for the motion described in the previous parts?
Explanation / Answer
Now the idea of average velocity is something that is fairly straightforward, but the idea of instantaneous velocity is a little trickier. It really requires calculus to fully appreciate, but hopefully you already know what a derivative is, so this shouldn't be too hard. Suppose the velocity of the car is varying, because for example, you're in a traffic jam. You look at the speedometer and it's varying a lot, all the way from zero to 60 mph. What is the instantaneous velocity? It is, more or less, what you read on the speedometer. I'm assuming you've got a good speedometer that isn't too sluggish and can change its reading quite quickly. Your speedometer is measuring the the average velocity but one measured over quite a short time, to ensure that you're getting an up to date reading of your velocity. So if you measure the displacement of the car over a time , you can use that to determine the average velocity of the car. What we want is to take the limit as goes to zero. More formally, the instantaneous velocity v is defined as Most of the time we'll be working with instantaneous velocity, so we'll just drop the instantaneous, and call the above v the velocity. To justify that such a limit exists is something that you've hopefully had to grapple with already. For physics problems, this limit does indeed exist and gives the derivative:
