Velocity

Definition of Velocity

Velocity is a vector quantity used to describe an object's direction of motion and speed. It is often characterized by two types, average velocity, and instantaneous velocity. Average velocity is a vector quantity that relies on the final and initial position of an object.

Instantaneous velocity is the velocity of an object at a specific moment in time.

Formula for Velocity

The mathematical formula corresponding to the definition of average velocity is

$$ v_{avg} = \frac{ \Delta x }{ \Delta t}, $$

where \( \Delta x \) is the displacement measured in meters \(( \mathrm{m} )\) and \( \Delta t \) is time measured in seconds \(( \mathrm{s} )\). Note that if we take the derivative of this, the equation becomes \( v = \frac{ \mathrm{d}x }{ \mathrm{d}t } \), where \( dx \) is are infinitely small change in displacement and \( dt \) is are infinitely small change in time. If we let time go to zero, this equation now gives us the mathematical formula corresponding to the definition of instantaneous velocity.

One can also calculate the average velocity over time using the initial and final values of velocity.

$$v_{\text{avg}}=\frac{v_o + v}{2}$$

where \( v_o \) is initial velocity and \( v \) is final velocity.

This equation is derivable from the kinematic equation for average distance as follows:

$$\begin{aligned}\Delta{x}=& \frac{v_o+v}{2}(t) \\ \frac{\Delta{x}}{t}= & \frac{v_o+v}{2} \\ v_{\text{avg}}= & \frac{v_o+v}{2}. \\ \end{aligned}$$

SI Unit of Velocity

Using the formula for velocity, its SI unit is calculated as follows:

$$ v_{\text{avg}}= \frac{ \Delta x }{ \Delta t } = \frac{ \mathrm{m} }{ \mathrm{s} } $$

Therefore, the SI unit for velocity is \( \frac{ \mathrm{m} } { \mathrm{s} } \).

Calculating Average Velocity from an Acceleration-Time Graph

Another way to calculate average velocity over time is by means of an acceleration-time graph. When looking at an acceleration-time graph, you can determine the velocity of the object as the area under the acceleration curve is the change in velocity.

$$\text{Area}=\Delta{v}.$$

For example, the acceleration-time graph below represents the function, \( a(t)=0.5t+5 \) between \(0\,\mathrm{s}\) to \(5\,\mathrm{s}\). Using this, we can show that the change in velocity corresponds to the area under the curve.

Figure 2: Determining average velocity from an acceleration-time graph.

Using this graph, we can find what the velocity will be after a specific amount of time by understanding that the change in velocity is the integral of acceleration

$$\Delta v=\int_{t_1}^{t_2}a(t)$$

where the integral of acceleration is the area under the curve and represents the change in velocity. Therefore,

$$\begin{aligned}\Delta v&=\int_{t_1}^{t_2}a(t) \\ \Delta v&=\int_{t_1=0}^{t_2=5}(0.5t +5)dt\\ \Delta v&=\frac{0.5t^2}{2}+5t \\ \Delta v&=\left(\frac{0.5(5)^2}{2}+5(5)\right)-\left(\frac{0.5(0)^2}{2}+5(0)\right)\\ \Delta v&=31.25\,\mathrm{\frac{m}{s}}.\\\end{aligned}$$

We can double-check this result by calculating the area of two different shapes (a triangle and a rectangle) as the first figure shows.

Start by calculating the area of the blue rectangle:

$$\begin{aligned}\text{Area}&=(\text{height})(\text{width})=hw \\\text{Area}&=(5)(5)\\ \text{Area}&=25.\\\end{aligned}$$

Now calculate the area of the green triangle:

$$\begin{aligned}\text{Area}&=\frac{1}{2}\left(\text{base}\right)\left(\text{height}\right)=\frac{1}{2}bh \\\text{Area}&=\frac{1}{2}\left(5\right)\left(2.5\right)\\ \text{Area}&=6.25.\\\end{aligned}$$

Now, adding these two together, we retrieve the result for the area under the curve:

$$\begin{aligned}\text{Area}_{\text{(curve)}}&=\text{Area}_{(\text{rec})}+ \text{Area}_{(\text{tri})} \\{Area}_{(\text{curve})}&= 25 + 6.25\\ \text{Area}_{(\text{curve})}&=31.25.\\\end{aligned}$$

The values match clearly, showing that in the acceleration-time graph, the area under the curve represents the change in velocity.

Instantaneous Velocity from a Graph

We can calculate average velocity and instantaneous velocity by means of a position-time graph and a velocity-time graph. Let's familiarize ourselves with this technique, starting with the velocity-time graph below.

Figure 3: A velocity-time graph depicting constant velocity.

From this velocity-time graph, we can see that the velocity is constant with respect to time. Consequently, this tells us that the average velocity and the instantaneous velocity are equal because velocity is constant. However, this is not always the case.

Figure 4: A velocity-time graph depicting a scenario when velocity is not constant with respect to time.

When looking at this velocity-time graph, we can see that the velocity is not constant as it is different at different points. This tells us that average velocity and instantaneous velocity are not equal. However, to better understand instantaneous velocity, let's use the position-time graph below.

Figure 5: A position-time graph depicting instantaneous velocity as slope.

Suppose the blue line on the graph above represents a displacement function. Now using the two points seen on the graph, we could find the average velocity by using the equation, \( v_{avg}=\frac{\Delta{x}}{\Delta{t}} \) which is simply the slope between those points. However, what will happen if we make one point a fixed point and vary the other, so it gradually approaches the fixed point? In simple terms, what will happen as we make the change in time smaller and smaller? Well, the answer is instantaneous velocity. If we vary one point, we will see that as the time approaches zero, the time interval becomes smaller and smaller. Therefore, the slope between these two points becomes closer and closer to the line tangent at the fixed point. Hence, the line tangent to the point is in fact instantaneous velocity.

Difference Between Velocity and Speed

In everyday language, people often consider the words velocity and speed as synonyms. However, although both words refer to an object's change in position relative to time, we consider them as two distinctly different terms in physics. To distinguish one from the other, one must understand these 4 key points for each term.

Speed corresponds to how fast an object is moving, accounts for the entire distance an object covers within a given time period, is a scalar quantity, and cannot be zero.

Velocity corresponds to speed with direction, only accounts for an object's starting position and final position within a given time period, is a vector quantity, and can be zero. Their corresponding formulas are as follows:

\begin{aligned} \mathrm{Speed} &= \mathrm{\frac{Total\,Distance}{Time}} \\ \mathrm{Velocity} &= \mathrm{\frac{Displacement}{Time} = \frac{Final\,Position - Starting\,Position}{Time}}.\end{aligned}

A simple way to think about speed and velocity is walking. Let's say you walk to the corner of your street at \( 2\,\mathrm{\frac{m}{s}} \). This only indicates speed because there is no direction. However, if you go north \( 2\,\mathrm{\frac{m}{s}} \) to the corner, then this represents velocity, since it includes direction.

Instantaneous Velocity and Instantaneous Speed

When defining speed and velocity, it is also important to understand the concepts of instantaneous velocity and instantaneous speed. Instantaneous velocity and instantaneous speed both are defined as the speed of an object at a specific moment in time. However, the definition of instantaneous velocity also includes the object's direction. To better understand this, let us consider an example of a track runner. A track runner running a 1000 m race will have changes in their speed at specific moments in time throughout the entire race. These changes might be most noticeable toward the end of the race, the last 100 m, when runners begin to increase their speed to cross the finish line first. At this particular point, we could calculate the instantaneous speed and instantaneous velocity of the runner and these values would probably be higher than the runner's calculated speed and velocity over the entire 1000m race.

Velocity Example Problems

When solving velocity problems, one must apply the equation for velocity. Therefore, since we have defined velocity and discussed its relation to speed, let us work through some examples to gain familiarity with using the equations. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to help illustrate what is happening and provide a visual aid for yourself.

Examples

Let's use our newfound knowledge of velocity to complete some examples involving average velocity and instantaneous velocity.

Now, lets complete a slightly more difficult example that will involve some calculus.