Physics Tutorial: Speed and Velocity in 1 Dimension

[ 1 Votes ]
  • The meaning of Speed
  • How to calculate the speed of a moving object?
  • The difference between average and instantaneous speed
  • The meaning of velocity and how does it differs from speed
  • How to calculate the velocity of a moving object?
  • The difference between average and instantaneous velocity
  • Expressing the velocity as the gradient (slope) of the Position vs Time graph

    Introduction

    In the previous two tutorials "Displacement and Distance in One Dimension" and "Displacement and Distance in Two and Three Dimensions", we discussed extensively about these terms as concepts related to the motion of objects.

    Do you think it is enough to know the Displacement and Distance of an object to have an idea on how the object is moving? Why?

    Many people use the terms "Speed" and "Velocity" without making any distinction between them. They consider these terms as synonyms. But are they really so?

    In this tutorial, we explain the meaning of these concepts using a very basic terminology, so the readers will understand easier the abovementioned concepts and the difference between them.

    What is Speed? How do we measure it?

    Let's try to give an answer to the first question posed in the "Introduction" part through an example.

    Eight athletes are participating in a 400 m running race. All of them manage to complete the race. Do you think they are awarded equally by the end of the race? Why?

    Obviously, the answer is NO. Each race has its own winners. Sometimes it is enough to complete a task to be the winner (in marathon or rally races) but this is not our case. Since all participants (athletes) are able to complete their race, the jury must use another criterion to make the distinction between the participants and not simply to evaluate the race completion. This criterion consists on completing the race in the shortest time possible. Therefore, a new quantity is introduced here to determine the winner in addition to the running distance. This quantity is the "Time of Motion" (in short Time) which is denoted by "t" and is measured in seconds [s].

    When combining the moving distance and time we obtain a new Kinematic quantity known as Speed (v).

    By definition, Speed (v) is the Distance travelled by an object in a given time (sometimes we say "in the unit of time" instead of "in a given time").

    Mathematically, we can write:

    v = s/t

    Since (in the SI system of units) the Displacement is measured in metres and Time in seconds, the unit of speed is

    Unit of speed = [metre/second] = [m/s]

    For example, if the winner completes the 400 m race in 80 seconds, his speed is

    v = s/t
    = 400m/80s
    = 5 m/s

    Since Speed is obtained by dividing two scalar quantities such as Distance and Time, it is a scalar quantity as well. Therefore, no direction is involved when discussing about moving speed of an object.

    The meaning of Average and Instantaneous Speed

    In the previous example, we found the speed of winner athlete was 5 m/s. It is obvious this represents an average value, as it is impossible for the athlete to keep moving at this speed during the entire race. Normally an athlete runs faster at the beginning of a race and then he slows down, as he gets tired. Therefore, we use the concept of "average speed" to represent a motion, which is presumably irregular.

    By definition, "The average speed < v > of an object refers to the total distance it travels divided by the time elapsed."

    Mathematically, we write

    < v > = stotal/ttotal

    The utility of the average speed use is better observed in the following example.

    Example 1

    Physics Tutorials: This image shows

    What is the average speed of the car (in km/h) during the entire trip?

    Solution 1

    It is obvious the motion has not been uniform during the trip. Therefore, we must use the concept of average speed to understand whether the car has moved fast or slowly during the process.

    From the equation of the average speed

    < v > = stotal/ttotal

    and relying on the data shown in the figure, we see that

    < v > = sAB + sBC + sCD + sDE/tAB + tBC + tCD + tDE

    Substituting the values, we obtain for the average speed of the car

    < v > = 70km + 100km + 60km + 90km/1h + 2h + 0.5h + 1.5h
    = 320km/5h
    = 64 km/h

    On the other hand, Instantaneous Speed (v) is another important concept used in Kinematics. It shows the actual speed at which an object is moving. An example of application in daily life of the instantaneous speed concept is the car's speedometer. It shows the instantaneous speed of the car as it takes "live" measurements of the car's moving speed.

    But how can we measure the instantaneous speed when an object is not doing a uniform motion and furthermore when its motion is quite irregular?

    The procedure is not very simple. However, with the help of digital electronics this is a solved issue already. Basically, a very sensitive timer is used to partialize the motion in small intervals and for each interval the distance is calculated through a distance sensor. The speed cannot change drastically due to the object's "inertia" (look at the tutorial Newton's First Law of Motion. The Meaning of Inertia). Therefore, the average speed measured for each of the abovementioned small intervals of time represents the instantaneous speed of the object in that specific instant. The result obtained is accurate enough as any change in values is smooth considering the shortness of time intervals.

    To calculate instantaneous speed, we need to divide the distance by time. However, we don't want to use the distance of the entire trip, because that will give us average speed. We take a small portion of distance and divide it by the corresponding (small) time interval. The smaller the distance (and the time interval) used, the more accurately we can measure the speed for that specific time.

    The equation of instantaneous speed v is:

    v = ∆s/∆t

    where Δs represents the small distance considered and Δt the time interval during which the event occurs.

    If it is possible to represent the motion in a graph this would be a huge advantage as the value of the instantaneous speed is calculated directly from the data shown in the graph. Look at the example below.

    Example 2

    The following graph shows the dependence of the Distance travelled by an object from the Time elapsed. Find the speed of the object at t = 4.4 s.

    Physics Tutorials: This image shows

    Solution 2

    As explained before, we must take two neighbouring values for the distance (s1 and s2) and the time (t1 and t2), which must be as close as possible and then using the equation

    v = ∆s/∆t= s2 - s1/t2 - t1

    to calculate the instantaneous speed at the required instant.

    Let's take for example t1 = 4.1 s and t2 = 4.6 s. The corresponding distances for these two time values are s1 = 15.0 m and s2 = 18.0 m respectively. Look at the figure:

    Physics Tutorials: This image shows

    Therefore, we obtain for the instantaneous speed at t = 4.4 s or as it is otherwise written, v (4.4):

    v = ∆s/∆t
    = s2-s1/t2-t1
    = 18.0m - 15.0m/4.6s - 4.1s
    = 3.0m/0.5s
    =6 m/s

    This is an approximate value, which becomes closer to the true value when the intervals become finer. However, it gives us a good idea on how the object is moving.

    What is Velocity? How do we measure it?

    As discussed above, Speed is the Distance travelled in the unit of Time. But in the previous tutorials we have learned that there is another quantity besides Speed which is measured in the units of length. This quantity was called Displacement and its shows the change in position of an object. So, what happens is we divide Displacement by Time? Do we still get Speed?

    The answer is NO. When dividing Displacement and Time, we obtain a new physical quantity known as Velocity.

    Velocity is a vector quantity and it is denoted by v. Its formula is:

    Velocity = Displacement/Time

    In symbols,

    v = ∆x/t

    (In this instance we are discussing Speed and Velocity in one dimension, so Displacement is written as ∆x instead of ∆r).

    Since Displacement is measured in metres and Time in seconds, the unit of Velocity is [m/s] just like the unit of Speed.

    It is obvious Velocity is a vector quantity as it is obtained by dividing a vector by a scalar (look at the Physics tutorial Multiplication of a Vector by a Scalar).

    Example

    An object moves uniformly for 20 seconds from A to C according the path shown in the figure.

    Physics Tutorials: This image shows

    Calculate:

    1. Total Distance
    2. Total Displacement
    3. Speed
    4. Velocity

    of the object. Take π = 22/7.

    Solution

    The object has moved in such a path that we have a half-circle and a straight line combined. For the half circle, we have:

    sAB = s1 = 1/2 of Circumference
    = 1/2 × 2 × π × r
    = π × r
    = 22/7 × 35 m
    = 110 m

    The Distance s2 is given; it is sBC = s2 = 80m.

    Therefore, the total Distance s is:

    s = s1 +s 2
    = 110m + 80m
    = 190m

    Displacement (as the shortest path from A to C), represents the hypotenuse of the right triangle formed by the diameter of the half-circle AB and the line BC as shown in the figure.

    Physics Tutorials: This image shows

    Thus, since AB = d = 2r = 2 × 35 m = 70 m, we have for the Displacement (The length of the vector AC):

    |AC|2 = |AB|2 + |BC|2
    |AC|2 = (70m)2 + (80m)2
    = 4900m2 + 6400m2
    = 11300m2

    Hence,

    |∆x| = |AC| = √11300m2
    ≈ 106m

    Now, it is easier to calculate the Speed. We simply divide the Distance found at (a) by the Time. Thus,

    v = s/t
    = 190m/20s
    = 9.5 m/s

    The same procedure is used to calculate the magnitude of Velocity, i.e. we will divide the Displacement found at (b) by the Time, t. Thus,

    |v| = |∆x|/t
    = 106m/20s
    = 5.3 m/s

    Average and Instantaneous Velocity

    The approach is the same as for Average and Instantaneous Speed. Thus, Average Velocity < v > represents the total Displacement divided by the total Time taken. The concept of average velocity is particularly useful when dealing with non-uniform motion.

    Mathematically, we have:

    < v > = ∆xtot/ttot

    As for Instantaneous Velocity v, again we take two neighbouring values for Position and Time. In this way, we obtain one interval for each quantity, namely ∆x and ∆t. Therefore, we obtain the equation

    v = ∆x/∆t

    Example

    Calculate the average velocity of the entire motion and the instantaneous velocity at t = 4.0 s for a moving object whose Position vs Time graph is shown in the figure below.

    Physics Tutorials: This image shows

    Solution

    From the graph, we see that the initial position is xi = 9.0m and the final position is xf = 2.0m, the initial time is ti = 0 and the final time is tf = 6.5s. Therefore, applying the formula of average velocity

    < v > = ∆xtot/ttot = xf - xi/tf - ti
    /

    we obtain after the substitutions

    < v > = 9.0m - 2.0m/6.5s - 0s
    = 7.0m/6.5s
    = 1.1 m/s

    (The result is written at one decimal place to fit the clues. Look at the tutorail Significant Figures and Their Importance).

    As for the instantaneous velocity at the instant required (at t = 4.0s), again we take two surrounding values for each quantity shown in the graph (two for position and two for the time). These values must be as close as possible to the point required. For example, we can choose t1 = 3.8s and t2 = 4.2s. From the graph, we can see that the corresponding values of position are x1 = 6.2m and x2 = 6.8m as shown below.

    Physics Tutorials: This image shows

    Using the equation of the instantaneous velocity

    v = ∆x/∆t

    we obtain after substituting the known values:

    v = x2-x1/t2-t1
    6.8m - 6.2m/4.2s - 3.8s
    = 0.6m/0.4s
    = 1.5 m/s

    Remark! From Mathematics, it is known that the slope of a graph (otherwise known as gradient) at a certain point is obtained by dividing the change in vertical coordinate to the change in the horizontal one for a small interval surrounding the given point. This can be observed in our example as well. Look at the magnified section of the graph around the instant t = 4.0 s.

    Physics Tutorials: This image shows

    Therefore, we obtain a very important rule in Kinematics:

    "The slope (gradient) of the Position vs Time graph gives the Velocity."

    This rule helps a lot in understanding the properties of velocity and gives it a mathematical meaning.

    Summary

    When combining the moving distance and time we obtain a Kinematic quantity known as Speed (v).

    By definition, Speed (v) is the Distance travelled by an object in a given time (sometimes we say "in the unit of time" instead of "in a given time").

    Mathematically, we can write

    v = s/t

    Since (in the SI system of units) the Displacement is measured in metres and Time in seconds, the unit of speed is

    Unit of speed = [metre/second] = [m/s]

    The average speed < v > of an object refers to the total distance it travels divided by the time elapsed.

    Mathematically, we write

    < v > = stotal/ttotal

    Speed is a scalar quantity as it is obtained by dividing two scalars: distance and time of motion.

    Instantaneous Speed (v) is another important concept used in Kinematics. It shows the actual speed by which an object is moving.

    To calculate the instantaneous speed of an object, we take a small portion of distance and divide it by the corresponding (small) time interval. The smaller the distance (and the time interval) used, the more accurately we can measure the speed for that specific time.

    The equation of instantaneous speed v is

    v = ∆s/∆t

    where Δs represents the small distance considered and Δt the time interval during which the event occurs.

    When dividing Displacement and Time, we obtain a new physical quantity known as Velocity.

    Velocity is a vector quantity and it is denoted by v. Its formula is

    Velocity = Displacement/Time

    In symbols (in one dimension) the above equation is written as

    v = ∆x/t

    Since Displacement is measured in metres and Time in seconds, the unit of Velocity is [m/s] just like the unit of Speed.

    It is obvious Velocity is a vector quantity as it is obtained by dividing a vector by a scalar.

    If we want to calculate the average and instantaneous velocity, the approach is the same as for average and instantaneous speed. Thus, Average Velocity < v > represents the total Displacement divided by the total Time taken. The concept of average velocity is particularly useful when dealing with non-uniform motion.

    Mathematically, we have:

    < v > = ∆xtot/ttot

    As for the Instantaneous velocity v, again we take two neighbouring values for Position and Time. In this way, we obtain one interval for each quantity, namely ∆x and ∆t. Therefore, we obtain the equation

    v = ∆x/∆t

    A very important rule in Kinematics states that:

    The slope (gradient) of the Position vs Time graph gives the Velocity.

    Speed and Velocity in 1 Dimension Revision Questions

    1. An object moves along the path shown in the figure below.

    Physics Tutorials: This image shows

    Calculate the Speed and the Velocity if it takes 8 s to the object to travel from A to B according the given path. Take π = 22/7.

    1. Speed = 22 m/s, Velocity = 15 m/s
    2. Speed = 176 m/s, Velocity = 56 m/s
    3. Speed = 15 m/s, Velocity = 22 m/s
    4. Speed = 44 m/s, Velocity = 30 m/s

    Correct Answer: A

    2. What is the instantaneous velocity at t = 5.0 s for the moving object whose Position vs Time graph is shown below?

    Physics Tutorials: This image shows
    1. 4 m/s
    2. - 4 m/s
    3. 2 m/s
    4. -2 m/s

    Correct Answer: B

    3. A car travels from the city A to the city B at 60 m/s and turns back from B to A at 120 m/s. What is the average speed of the entire trip?

    1. 120 m/s
    2. 90 m/s
    3. 80 m/s
    4. 60 m/s

    Correct Answer: C

    For more insight on Speed and Velocity, please see the next tutorial: "Displacement and Distance in 2 and 3 Dimensions."

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