Velocity Vector

A position vector describes the position of a point in a coordinate system.

Figure:7.a

If P is the position of an object at time t, then OP is the position vector of the object at that time. It is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>&#x2192;</mo></mover></math>.

Displacement vector describes the position of a point with reference to a point other than the origin of the coordinate system.

Figure:7.b

An object moves from a point P to point Q. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>r</mi><mn>1</mn></msub><mo>&#x2192;</mo></mover></math> is the position vector at time $t_{1}$ and  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>r</mi><mn>2</mn></msub><mo>&#x2192;</mo></mover></math> is the position vector at time $t_{2}$.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2206;</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></math> is the displacement vector directed from point P to Q, corresponding to the motion of the object from P to Q.

The average velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover></math> of an object is the ratio of the displacement and the corresponding time interval.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><mfrac><mrow><mo>&#x2206;</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac></math>

Or,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><mfrac><mrow><mo>&#x2206;</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mover><mi>i</mi><mo>^</mo></mover><mo>&#x2206;</mo><mi>x</mi><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mo>&#x2206;</mo><mi>y</mi></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mi>x</mi></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mi>y</mi></msub></math>

The instantaneous velocity is given by the limiting value of the average velocity as the time interval approaches zero.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><munder><mi>lim</mi><mrow><mo>&#x2206;</mo><mi>t</mi><mo>&#x2192;</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>&#x2206;</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>

The instantaneous velocity vector is always tangential to the path at the point that locates the position of the object.

Figure:7.c

Also, we have,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover><mo>=</mo><mfrac><mrow><mo>&#x2206;</mo><mover><mi>r</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mover><mi>i</mi><mo>^</mo></mover><mo>&#x2206;</mo><mi>x</mi><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mo>&#x2206;</mo><mi>y</mi></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mi>x</mi></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mi>y</mi></msub></math>

The magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>&#x2192;</mo></mover></math> is given by,

$\vert v\vert =\sqrt{v_{x}^{2}+v_{y}^{2}}$

And the direction is given by,

$\theta =\tan ^{-1}\frac{v_{y}}{v_{x}}$

Acceleration:

The average acceleration of an object for a time interval t is the change in velocity divided by the time interval.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>=</mo><mfrac><mrow><mo>&#x2206;</mo><mover><mi>v</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac></math>

Or,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>=</mo><mfrac><mrow><mo>&#x2206;</mo><mover><mi>v</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mover><mi>i</mi><mo>^</mo></mover><mo>&#x2206;</mo><msub><mi>v</mi><mi>x</mi></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mo>&#x2206;</mo><msub><mi>v</mi><mi>y</mi></msub></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>a</mi><mi>x</mi></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>a</mi><mi>y</mi></msub></math>

The instantaneous acceleration is given by the limiting value of the average acceleration as the time interval approaches zero. The instantaneous acceleration is shown at point P in the below diagram.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>=</mo><munder><mi>lim</mi><mrow><mo>&#x2206;</mo><mi>t</mi><mo>&#x2192;</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>&#x2206;</mo><mover><mi>v</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>&#x2206;</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mover><mi>v</mi><mo>&#x2192;</mo></mover></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>

Figure:7.d

As $\Delta t$ decreases, the direction of velocity changes and consequently, the direction of the acceleration changes. When $\Delta t$tends to zero, the average acceleration becomes the instantaneous acceleration and has the direction as shown in figure 7.d.

Instantaneous acceleration does not have the same direction as that of velocity vector.

Velocity and acceleration vectors may have any angle between $0^{0}$ and $180^{0}$between them.

Relative velocity in two dimensions:

Earlier we have studied that if two objects A and B are moving with uniform velocities $V_{A}$and $V_{B}$ along the same direction respectively, then:

relative velocity of A with respect to B is $V_{AB}=V_{A}-V_{B}$ ………….. (1)

And relative velocity of B with respect to A is $V_{BA}=V_{B}-V_{A}$ ………. (2)

Now components of velocity of object A along the X and Y-directions,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>v</mi><mi>A</mi></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>X</mi><mi>A</mi></mrow></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>Y</mi><mi>A</mi></mrow></msub></math>
……………. (3)

Components of velocity of object B along the X and Y-directions,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>v</mi><mi>B</mi></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>X</mi><mi>B</mi></mrow></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>Y</mi><mi>B</mi></mrow></msub></math>
…………….. (4)

From equation (1), we get,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>v</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>X</mi><mi>A</mi></mrow></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>Y</mi><mi>A</mi></mrow></msub><mo>-</mo><mfenced><mrow><mover><mi>i</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>X</mi><mi>B</mi></mrow></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>v</mi><mrow><mi>Y</mi><mi>B</mi></mrow></msub></mrow></mfenced></math>

Or,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>v</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><mfenced><mrow><msub><mi>v</mi><mrow><mi>X</mi><mi>A</mi></mrow></msub><mo>-</mo><msub><mi>v</mi><mrow><mi>X</mi><mi>B</mi></mrow></msub></mrow></mfenced><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mfenced><mrow><msub><mi>v</mi><mrow><mi>Y</mi><mi>A</mi></mrow></msub><mo>-</mo><msub><mi>v</mi><mrow><mi>Y</mi><mi>B</mi></mrow></msub></mrow></mfenced></math>

Similarly,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>v</mi><mrow><mi>B</mi><mi>A</mi></mrow></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><mfenced><mrow><msub><mi>v</mi><mrow><mi>X</mi><mi>B</mi></mrow></msub><mo>-</mo><msub><mi>v</mi><mrow><mi>X</mi><mi>A</mi></mrow></msub></mrow></mfenced><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mfenced><mrow><msub><mi>v</mi><mrow><mi>Y</mi><mi>B</mi></mrow></msub><mo>-</mo><msub><mi>v</mi><mrow><mi>Y</mi><mi>A</mi></mrow></msub></mrow></mfenced></math>

Conceptual question:

  1. Show that the relative speed between two objects moving in opposite directions is equal to the sum of their individual speeds.

We have,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></mfenced><mo>=</mo><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>B</mi><mi>A</mi></mrow></msub></mfenced><mo>=</mo><msqrt><msubsup><mi>V</mi><mi>A</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>V</mi><mi>B</mi><mn>2</mn></msubsup><mo>-</mo><mn>2</mn><msub><mi>V</mi><mi>A</mi></msub><msub><mi>V</mi><mi>B</mi></msub><mi>cos</mi><mi>&#x3B8;</mi></msqrt></math>

As the objects are moving in opposite directions, then $\theta =180^{0}$.

Therefore,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></mfenced><mo>=</mo><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>B</mi><mi>A</mi></mrow></msub></mfenced><mo>=</mo><msqrt><msubsup><mi>V</mi><mi>A</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>V</mi><mi>B</mi><mn>2</mn></msubsup><mo>-</mo><mn>2</mn><msub><mi>V</mi><mi>A</mi></msub><msub><mi>V</mi><mi>B</mi></msub><mi>cos</mi><msup><mn>180</mn><mn>0</mn></msup></msqrt><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><msqrt><msubsup><mi>V</mi><mi>A</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>V</mi><mi>B</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>V</mi><mi>A</mi></msub><msub><mi>V</mi><mi>B</mi></msub></msqrt><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msqrt><msup><mfenced><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>+</mo><msub><mi>V</mi><mi>B</mi></msub></mrow></mfenced><mn>2</mn></msup></msqrt><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xB1;</mo><mfenced><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>+</mo><msub><mi>V</mi><mi>B</mi></msub></mrow></mfenced></math>

As speed cannot be negative, therefore,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub></mfenced><mo>=</mo><mfenced open="|" close="|"><msub><mi>V</mi><mrow><mi>B</mi><mi>A</mi></mrow></msub></mfenced><mo>&#xA0;</mo><mo>=</mo><mfenced><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>+</mo><msub><mi>V</mi><mi>B</mi></msub></mrow></mfenced></math>

Thus, the relative speed between two objects moving in opposite directions is equal to the sum of their individual speeds.

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