Physics Vector Calculator

Introduction to physics vector calculator:

Vector calculation problems in physics are those problems that require the application of vectors and operations on vectors. Many complex calculator problems in physics are solved easily by using vectors. Since vectors are different from scalar quantities in that they have direction as well as magnitude, even the basic arithmetic operations on vectors are different from those on scalars.

Addition, subtraction and multiplication of vectors, all have different methods than those of simple algebra.

In this short tutorial, we deal with physics vector problems that are solved using these operations on vectors.

Physics Vector Application Problems and Solutions

A river flows at 3 m/s and is 300 m wide. A man swims across the river with a velocity of 2 m.s directed always perpendicular to the flow of current. Find the magnitude of the resultant velocity of the man under the effect of the stream?

Given :

Velocity of river `vecVr` = 3 m/s

Width of river, `W` = 300 m

Velocity of swimmer, `vecVs` = 2 m/s

Angle between velocity of river and velocity of swimmer, `theta` = 90°

The following diagram illustrates the given situation:

In the above diagram, velocity '`vecV` ' is the resultant velocity of the man under the influence of the stream.

Since the resultant velocity of the swimmer (`vecV` ) is the result of the combined velocities of the river and the swimmer, therefore

`vecV = vecVs + vecVr`

By applying the vector addition formula derived from triangle law of vector addition, we get

`vecV = sqrt((Vs)^2 + (Vr)^2 + (Vs)(Vr)*costheta)`

`vecV = sqrt((2)^2 + (3)^2 + (2)(3)*cos90)`

`vecV = sqrt(4 + 9)`

`vecV = sqrt(13) = 3.6 m/s`

Thus, the resultant velocity of the swimmer under the action of the stream is 3.6 m/s.

Explanation to above Physics Vector Application Problem

In the above question, find the direction of the resultant velocity `vecV` of the swimmer with respect to the shore of the river.

Let the angle of inclination of resultant velocity vector `vecV` to the velocity of the river `vecVr` be `alpha` °.

We know that angle between velocity of river and that of swimmer, `theta = 90` °

Thus applying the formula for velocity of resultant vector,

`tan(alpha) = (Vb sintheta)/(Va - Vbcostheta)` , where `Vb` is the velocity of swimmer and `Va` the velocity of river.

`tan(alpha) = (Vs sintheta)/(Vr - Vscostheta)`

`tan(alpha) = (2 sin90)/(3 - 2cos90)`

`tan(alpha) = 2/3 = 0.66`

By referring to the trigonometric table for tangent, we get `alpha = 33.7` ° (approx).

Thus, the velocity of the swimmer has a direction of 33.7° inclination to the shore of the river.