- Vectors in Physics
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- Two Dimensional Motion
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- Motion in a plane with constant acceleration
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- Relative Velocity in two dimension
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- Projectile Motion
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- Uniform circular motion
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- Motion in three dimensions
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- Sample Problems with Solutions

In the coordinate system, the line joining the origin O to the point P in the system is called the position vector of Point P

Let Point P and Q are there

Position Vector of Point P

Position Vector of Point Q

=(x

- Consider a particle moving along a curved path in x-y plane shown below in the figure

- Suppose at any time,particle is at the point P and after some time 't' is at point Q where points P and Q represents the position of particle at two different points.

- Position of particle at point P is described by the Position vector
**r**from origin O to P given by

**r**=x**i**+y**j**

where x and y are components of**r**along x and y axis

- As particle moves from P to Q,its displacement would be would be Δ
**r**which is equal to the difference in position vectors**r**and**r'**.Thus

Δ**r**=**r'**-**r**= (x'**i**+y'**j**)-(x**i**+y**j**) = (x'-x)**i**+(y'-y)**j**= Δx**i**+Δy**j**(1)

where Δx=(x'-x) and Δy=(y'-y)

- If Δt is the time interval during which the particle moves from point P to Q along the curved path then average velocity(
**v**_{avg}) of particle is the ratio of displacement and corresponding time interval

since**v**_{avg}=Δ**r**/Δt , the direction of average velocity is same as that of Δ**r**

- Magnitude of Δ
**r**is always the straight line distance from P to Q regardless of any shape of actual path taken by the particle.

- Hence average velocity of particle from point P to Q in time interval Δt would be same for any path taken by the particle.

- We already know that instantaneous velocity is the velocity of the particle at any instant of time or at any point of its path.

- If we bring point Q more and more closer to point P and then calculate average velocity over such a short displacement and time interval then

where**v**is known as the instantaneous velocity of the particle.

- Thus, instantaneous velocity is the limiting value of average velocity as the time interval approaches zero.

- As the point Q approaches P, direction of vector Δ
**r**changes and approaches to the direction of the tangent to the path at point P. So instantaneous vector at any point is tangent to the path at that point.

- Figure below shows the direction of instantaneous velocity at point P.

- Thus, direction of instantaneous velocity
**v**at any point is always tangent to the path of particle at that point.

- Like average velocity we can also express instantaneous velocity in component form

where v_{x}and v_{y}are x and y components of instantaneous velocity.

- Magnitude of instantaneous velocity is

|**v**|=√[(v_{x})^{2}+(v_{y})^{2}]

and angle θ which velocity vector makes with x-axis is

tanθ=v_{x}/v_{y}

- Expression for instantaneous velocity is

Thus, if expression for the co-ordinates x and y are known as function of time then we can use equations derived above to find x and y components of velocity.

- Suppose a particle moves from point P to point Q in x-y plane as shown below in the figure

- Suppose
**v**_{1}is the velocity of the particle at point P and**v**_{2}is the velocity of particle at point Q

- Average acceleration is the change in velocity of particle from
**v**_{1}to**v**_{2}in time interval Δt as particle moves from point P to Q. Thus average acceleration is

Average acceleration is the vector quantity having direction same as that of Δ**v**.

- Again if point Q approaches point P, then limiting value of average acceleration as time approaches zero defines instantaneous acceleration or simply the acceleration of particle at that point. This, instantaneous acceleration is

- Figure below shows instantaneous acceleration a at point P.

- Instantaneous acceleration does not have same direction as that of velocity vector instead it must lie on the concave side of the curved surface.

- Thus velocity and acceleration vectors may have any angle between 0 to 180 degree between them.

The position of a object is given by

Where t is in second and coefficients have the proper units for r to be in centimetres

a) Find v(t) and a(t) of the object

b) Find the magnitude and direction of the velocity at t=3 sec

It is given in the questions

Now

Therefore,

Now

Therefore,

So acceleration is 4 cm/s

Now velocity at 3 sec

So its magnitude is √(3

And direction will be tan

Class 11 Maths Class 11 Physics Class 11 Chemistry