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Resultant Velocity Calculator

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What is velocity?What is resultant velocity? Velocity in 2DHow to find the resultant velocity: ExampleFAQs

This resultant velocity calculator is an Omni Calculator tool that allows you to add up to five different two-dimensional velocity vectors. It will instantaneously give you the magnitude of the resultant velocity, as well as its direction and its x and y components.

In the text below, we will explain the definition of velocity, what the resultant velocity is, how to add vectors with two components, and how to find the resultant velocity.

So grab your helmet, fasten your seat belt, and let's start to add some velocity vectors 🏎️!

What is velocity?

Usually, we are in motion. When you walk on the streets, play soccer with your friends on Sunday, go to the supermarket, drive your kids to school, ride your bicycle, etc. The common feature behind all these examples of motion is that in doing them, we cover a displacement in a given amount of time. The measure of how fast a displacement happens is called velocity. Such a relation can be mathematically described as:

vav=ΔrΔt\footnotesize \vec{v}_{\rm av} = \frac{\Delta \vec{r}}{\Delta t}

where

  • vav\vec{v}_{\rm av} — Average velocity;
  • Δr\Delta \vec{r} — Displacement vector; and
  • Δt \Delta t — Time interval.

Here, we can see that the average velocity is a vector. Therefore, it is characterized by an absolute value (also known as speed), and a direction. The average velocity can describe a motion where there is no acceleration, which means that the object is moving with constant speed. If you want to compute the velocity of an object subject to acceleration, then you need to use the definition of instantaneous velocity, whose formula is:

vinst=limΔt0vav=limΔt0ΔrΔt=drdt\footnotesize \begin{split} \vec{v}_{\rm inst} &= \lim_{\Delta t \rightarrow 0} \vec{v}_{\rm av} \\[1em] &= \lim_{\Delta t \rightarrow 0} \frac{\Delta \vec{r}}{\Delta t} \\[1em] & = \frac{d\,\vec{r}}{d\,t} \end{split}

The SI units to measure the absolute value of the velocity (or speed) are meters per second or m/s. However, in our daily activities, we can also find velocities measured in km/h, mph, km/min, etc. In our resultant velocity calculator, you can choose among different units for speed. You can also use our speed conversion calculator to change the units of your speed properly.

What is resultant velocity? Velocity in 2D

As we saw before, velocity is a vector with at least one component if we consider an object moving in one dimension. In our resultant velocity calculator, we will explore two-dimensional velocity vectors with components in x\rm{x} and y\rm{y} directions. This kind of velocity vector can be applied in the context of the projectile and relative motion. It can also be reduced to one-dimensional motion by taking one of its components equal to zero.

🙋 You can learn more about two-dimensional motion with our projectile motion calculator. Moreover, if your objects move with speeds close to the speed of the light, you can check our velocity addition calculator.

A 2D velocity vector can be written in terms of its vxv_{\rm x} and vyv_{\rm y} components through the formula:

v=vxi^+vyj^v=vcosθi^+vsinθj^\footnotesize \begin{split} \vec{v} &= v_{\rm x}\,\hat{i} + v_{\rm y}\,\hat{j} \\[1em] \vec{v} &= v\cdot \cos\theta\,\hat{i} + v\cdot \sin\theta\,\hat{j} \end{split}

where θ\theta is the angle between v\vec{v} and the x-axis. The resultant velocity is, as the name says, the sum of different velocity vectors acting over an object, which means that

vres=v1+v2+v3++vn\footnotesize \vec{v}_{\rm res} = \vec{v}_1+\vec{v}_2+\vec{v}_3+\ldots+\vec{v}_n

Then, if we want to compute the resultant velocity, all we have to do is to determine the x\rm{x} and y\rm{y} components of each velocity vector. This is done in such a way that our formula is rewritten as:

vres=vxresi^+vyresj^vxres=v1cosθ1+v2cosθ2+vncosθnvyres=v1sinθ1+v2sinθ2+vnsinθn\footnotesize \begin{split} \vec{v}_{\rm res} &= v_{\rm x\,res}\,\hat{i} + v_{\rm y\,res}\,\hat{j} \\[1em] v_{\rm x\,res} &= v_1 \cdot \cos\theta_1 + v_2 \cdot \cos\theta_2 \\[1em] &\quad+ \ldots v_n \cdot \cos\theta_n \\[1em] v_{\rm y\,res} &= v_1 \cdot \sin \theta_1 + v_2 \cdot \sin\theta_2 \\[1em] &\quad+ \ldots v_n \cdot \sin\theta_n \\[1em] \end{split}

where:

  • viv_i — Absolute values of each one of the nn velocities;
  • θi\theta_i — Directions in respect to the x-axis of each one of the nn velocities;
  • vxresv_{\rm x\,res} — x component of the resultant velocity; and
  • vyresv_{\rm y\,res} — y component of the resultant velocity.

In our resultant velocity calculator, you can add up to five different two-dimensional velocity vectors. To sum more than two velocities, click on the checkbox next to the question "Do you want to add up to 5 velocities together?" and substitute the 0's in the extra velocities fields of the calculator, as required.

Moreover, vxresv_{\rm x\,res} and vyresv_{\rm y\,res} can be used to calculate the absolute value of vres\vec{v}_{\rm res} and also its direction. You can determine these quantities by checking that

vres=vresvresvres=vxres2+vyres2\footnotesize \begin{split} v_{\rm res} &= \sqrt{\vec{v}_{\rm res} \cdot \vec{v}_{\rm res}} \\[1em] v_{\rm res} &= \sqrt{v_{\rm x\,res}^2 + v_{\rm y\,res}^2} \end{split}
θres=arctan(vyresvxres)\footnotesize \theta_{\rm res} = \arctan\left(\frac{v_{\rm y\,res}}{v_{\rm x\,res}}\right)

where:

  • vresv_{\rm res} — Absolute value of the resultant velocity; and
  • θres\theta_{\rm res} — Direction of the resultant velocity.

Now that you know the theory, let us show you how to use our calculator with an example.

How to find the resultant velocity: Example

Imagine you are on the side of a river, seeing your friend crossing it inside a boat. The boat velocity relative to the water is 15km/h15 \, \text{km/h} pointing straight in your direction. However, suppose that the river has a current and the water moves with a constant and uniform speed of 7km/h7 \, \text{km/h}, perpendicular to the boat direction.

In this case, what will be the resultant velocity of the boat that you observe? And what will be its direction? Let us use our resultant velocity calculator to find the answers.

You can start by selecting your units properly and typing the boat velocity in the Velocity 1 field of our calculator. Then, we can consider 0 in the Angle 1 field since the boat is moving in your direction. Now, insert the water velocity in the next field of the calculator, and here, we must use 90°90\degree as the Angle 2 parameter since the river current is perpendicular to the boat direction.

Then you can see your answers appearing instantaneously, so fast as a Formula One car. Our calculator finds that vres=16.55km/hv_{\rm res} = 16.55\, \text{km/h} and θres=25°\theta_{\rm res} = 25\degree. It also shows the absolute values of the x and y components of the resultant velocity vector.

You can check your results using the formulas below:

vxres=15km/hvyres=7km/h\footnotesize \begin{split} v_{\rm x\,res} & = 15 \, \text{km/h} \\[.6em] v_{\rm y\,res} & = 7 \, \text{km/h} \end{split}
vres=152+7216.55km/hθres=arctan(715)25°\footnotesize \begin{split} v_{\rm res} & = \sqrt{15^2+7^2} \approx 16.55\, \text{km/h} \\[1em] \theta_{\rm res} &= \arctan \left(\frac{7}{15}\right) \approx 25\degree \end{split}
FAQs

What is the difference between resultant velocity and velocity?

The velocity represents the rate of change of the displacement of an object in a given frame. It is a vector quantity, meaning it is described by its magnitude and direction. The resultant velocity is the result of adding several velocity vectors acting on a given object simultaneously.

How do I calculate the resultant velocity?

Suppose that you have two velocity vectors with speeds v1 = 3 m/s and v2 = 6 m/s, and directions θ1 = 30 deg and θ2 = 60 deg. You can find the resultant velocity following the steps below:

  1. Calculate the vx res component using the equation:

    vx res = v1 cos θ1 + v2 cos θ2 ≈ 5.6 m/s

  2. Determine the vy res component through the formula:

    vy res = v1 sin θ1 + v2 sin θ2 ≈ 6.7 m/s

  3. Use the equations below to find the absolute value of the resultant velocity and its direction:

    vres = √ (vx res2 + vy res2) ≈ 8.7 m/s

    θres = arctan(vy res/vx res) ≈ 50 deg

  4. Or simply input your data in our resultant velocity calculator and save your time!

What is resultant velocity and average velocity?

The resultant velocity is determined by adding several velocity vectors acting over a given object. The average velocity is defined as an object's average rate of displacement. The absolute value of the average velocity is constant, while the magnitude of the resultant velocity may change if one or more of its components are time-dependent.

Can the resultant velocity be equal to zero?

Yes, we can have scenarios where the resultant velocity is equal to zero. This happens, for instance, if we have two velocity vectors with the same speed pointing in opposite directions. If you want to see this, follow the steps below:

  1. Take v1 = v2 = 2 m/s and θ1 = π/6 and θ2 = 7 π/6.

  2. Calculate the vx res and vy res component through the formulas:

    vx res = 2 cos(π/6) + 2 cos(7π/6) = 0

    vy res = 2 sin(π/6) + 2 sin(7π/6) = 0

  3. Substitute the results in the formula to the absolute value of the resultant velocity:

    vres = √( vx res2 + vy res2) = 0

  4. That is it! You proved that by adding two opposite vectors of the same magnitude, the result is zero.

Velocity 1

Input the 1st and 2nd velocities to compute the resultant velocity, its direction, and components in the x and y axes.

Velocity 2

Input the 2nd velocity and its direction.

Enjoy the absolute value of the resultant velocity, its direction, and the x and y components.

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