Vector and Motion | Physics, Type, facts, Simple explanation | aurayne

If the position of a point on an object in the plane or space is changing it is said to be in motion otherwise in rest. To study motion of any object, we have to go through the basics of vector analysis.

Vector Analysis 

Physical quantities are of two types. 

1. Scalar 

2 Vector 

Scalar Quantities 

The quantities which have only magnitude, not direction are called scalar quantities, for example, mass, distance, time, speed, volume, density, pressure, work, energy, power etc. 

Vector Quantities 

The quantities which possess both magnitude and direction and which obey the laws of vector addition are called vectors, for example displacement, velocity, force etc. 

Some Important Definitions Regarding Vectors 

(i) Equal Vectors  These are the vectors having same magnitude and direction. regardless of their initial positions. 

As shown in the figure, the vectors A̅ and B representing a quantity are equal vectors.

(ii) Opposite Vectors  Two parallel vectors which have equal magnitude but opposite directions are called opposite vectors.

As shown in the figure, C̅ and D̅ are opposite vectors. 

(iii) Unit Vector  A vector whose magnitude is equal to unity is called unit vector. If A̅ is a vector, whose magnitude is |A̅|,then |A̅| 

|A̅|is a unit vector. The direction of this vector is in the same direction of A̅.

The unit vector is denoted as Â. If we show a unit vector in particular direction, then a vector of desired magnitude can be obtained by multiplying the unit vector with the magnitude. 

A =  A̅

Orthogonal Unit Vectors  If î, ĵ and k̂ are assumed to be unit vectors along x, y and z-axes respectively, then a vector of magnitude 5 along x-axis can be expressed by 5 î.

(iv) Null Vector  It is a vector whose magnitude is zero and direction is undetermined.

Addition of Vectors 

It can be done by following two laws. These are triangle law of vector addition and parallelogram law of vector addition. 

Triangle Law of Vector Addition 

It states that, if a̅ and b̅ are two vectors to be added, then as shown in the diagram given below i.e, the tail of b̅ coincides with the head of a̅, the resultant will be a vector joining the tail of a̅ with the head of b̅.

⇒     AB + BC - AC

|AC|= √|AB|² + |BC|²  2·AB·BC cosθ 

Parallelogram Law of Addition of Vectors 

If two vectors acting at a point are represented both in magnitude and direction by the two adjacent sides of parallelogram drawn through that point then the diagonal passing through that point represents the resultant of those two vectors both in magnitude and direction. If θ be the angle between two vectors A̅ and B, then 

|R̅|= √A² + B² + 2AB cos θ 

and,      tan α = B sin θ / A+B cos θ' 

where α is the angle made by the resultant with vector A̅. 

The magnitude of the resultant varies between A̅ -|B̅ and  A̅ +|B̅  i.e.,

R̅max  = |A̅ + B̅  and  |R̅max| =  |A̅ - B̅|

Product of Two Vectors 

Multiplication of two vectors does not follows simple mathematical rule. These follows some special rules.

 

Scalar Product of Two Vectors 

The scalar product of two vectors is equal to the product of the magnitudes of the vectors and cosine of angle between them.

• Thus, if θ is the angle between two 

vectors A̅ and B̅, then the scalar product  of A̅ and B̅ will be

 A̅·B̅ = A B cosθ, 

where A and B are the magnitudes of vectors A̅ and B̅ respectively. 

Properties of Scalar Product 

The properties of scalar product are as follows:

(i) The scalar product of any vector with the same vector is equal to the square of the magnitude of that vector. 

A̅ · A̅ = A · A cos0 = A² 

(ii) The relation between the scalar products of orthogonal unit vectors î, ĵ and k̂ are as follows 

(a)     î · ĵ = ĵ · k̂ = k̂ · î = 0

(b)     î · î = ĵ · ĵ = k̂ · k̂ = 1

If vectors î, ĵ and k̂ each having magnitude 1 and angle between any two vectors is 90°, then,

î · ĵ = (1) · (1) cos90° = 0 

Similarly,  ĵ · k̂ = 0 and  k̂ · ĵ = 0

Again         î · î = (1) (1) cos 0 = 1 

Similarly,  ĵ · ĵ = 1 and k̂ · k̂ = 1

(iii) The scalar product of two vectors is equal to the sum of the product of the corresponding components of x, y and z.

Let, A̅ and B̅ be two vectors. Writing them in rectangular components, 

A̅ = AX î + Ay ĵ + AZ k̂ 

and,     B̅ = Bx î + By ĵ + Bz k̂ 

Hence, the angle between A̅ and B̅ is 

cos θ = A̅ · B̅ / AB 

= Ax Bx + Ay By + Az Bz / √A²x + A²y + A²z · √B²x + B²y  + B²z

Vector Product of Two Vectors 

The vector product of two vectors is the product of the magnitudes of those vectors and sine of angle between them and its direction is perpendicular to the plane of both vectors. 

If two vectors are A̅ and B̅ and the angle between A̅ and B̅ is θ, then the vector product will be 

A̅ × B̅   AB sin θ  n^

where A and B are the magnitudes of A̅ and B̅ respectively and n^ is a unit vector. 

which is perpendicular to the plane of A̅ and B̅.

Properties of Vector Product 

The properties of vector product are as follows 

(i) The vector product of any vector with the same vector is a zero vector. 

A̅ × A̅ = AA sin 0 n^ = 0 or 0 

(ii) The relation between the vector products of orthogonal unit î, ĵ and k̂ are as follows

(a)     î × ĵ = - ĵ × î = k̂

          ĵ × k̂ = - k̂ × ĵ = î

          k̂ × î = - î × k̂ = ĵ

It is essential to remember cyclic order in the above relation. 

(b)     î × î = ĵ × ĵ = k̂ × k̂ = 0 

(iii) The vector product of two vectors can be expressed in the form of determinant in terms of their corresponding components x, y and z. 

Let A̅ and B̅ be two vectors. Writing them in rectangular components. 

A̅ = Ax î + Ay  ĵ + Az k̂

B̅  =Bx î + By ĵ + Bz k̂

A̅ × B̅ =  

  

Motion 

If a body continuously changing its position with respect to a fixed point, then the body is said to be in motion. We study two types of motion. 

(a) Motion in one dimension 

(b) Motion in two and three dimensions 

Motion in One Dimension 

Some definitions related with motion 

(i) Distance  The length of total path covered by a body in any time is called distance. It is a scalar quantity i.e., does not depend on direction. 

(ii) Displacement The minimum distance between the initial and final position of a body is called the displacement of the body. It is a vector quantity i.e., depends on direction.  

(iii) Speed  The distance covered by a moving body in unit time is called the speed of the body. It is a scalar quantity. 

Speed = distance / time 

Average Speed  If a body covers different distances with different speed, then its average speed can be found with the formula. 

 Average speed = total distance / total time

If a body covers first half of a distance with speed v1 and second half with speed v2 then 

Average speed =  2 v1v2 / v1 + v2 

If a body covers first one-third distance with speed 'a', other one-third distance with speed 'b' and last one-third distance with speed 'c', then 

 Average speed = 3abc / ab + bc + ca 

Instantaneous Speed  The speed of the body at a given instant is called instantaneous speed. 

If Δt→0, then instantaneous speed 

 = limΔt→0  Δs/Δt = ds/dt

(iv) Velocity The rate of change of displacement of a body is called its velocity. It is a vector quantity, it is represented by 'v' and its unit is m/s.  

Average velocity = Total displacement/Total time        

Instantaneous Velocity The velocity of the body at a given instant is called instantaneous velocity. The velocity at given instant Vins = dr̅/dt

If a body is moving on a circular path, then after completing one complete cycle, its average velocity is zero. 

When a body covers equal half distances with the velocities v1 and v2 respectively, then 

Average speed =  2 v1v2 / v1 + v2 

When a body travels with velocity v1, for time t1 and v2 for time t2 then, 

Average velocity = v1 t1 + v2 t2  /  t1 + t2

(v) Acceleration  Rate of change of velocity is called acceleration. It is a vector quantity, it is represented by 'a' and its unit is 

a = Δv / Δt

If velocity of a body is decreasing, then the acceleration is called retardation or deceleration. 

Relative Velocity If the objects A and B moves with velocity vA and vB with respect to one coordinate system along a straight line in same direction, then velocity of B with respect to A will be v = vB  -   vA .  

Equations of Motion

Let a body starts to move with velocity u, after time t and after covering a distance s, velocity becomes v and acceleration of the body is a then there are three relations between these quantities, these relations are called equations of motion. 

(i) v = u + at

(ii) s = ut +  ½ at²

(iii) v² = u² + 2as 

Distance travelled in nth second is,

sn  = u + ½ a(2n - 1)

Special Cases 

(i) If a body starts from rest, then equations of motion are converted into 

(a) v = at

(b) s = ½ at²

(c) v² = 2as 

(ii) If velocity of body is decreasing in place of increasing, then equations of motion are converted into 

(a) v = u - at

(b) s = ut - ½ at²

(c) v² = u² - 2as

(iii) If the body is moving with uniform velocity, then equation of motion converted into 

(a) v = u

(b) s = ut

(c) v² = u²

Motion Under Gravity 

If a body is thrown upwards or falling downwards, then its motion is called motion under gravity. To obtain these equations, we put h (height) in place of s (distance), g (acceleration due to gravity) in place of a (acceleration). 

For upward motion 

(i) v = u - gt 

(ii) h = ut - ½ gt²

(iii) v² = u² - 2gh

For downward motion 

(i) v = u + gt 

(ii) h = ut + ½ gt²

(iii) v² = u² + 2gh

Expert TIPS

• If a body is thrown horizontally from some height and another body is dropped from same point, then time taken in both the cases are equal (provided resistance of air negligible). 

• Velocity and acceleration of a body may be in differe directions. 

• Velocity and acceleration of a body need. not to be zero simultaneously. 

• When, a body is dropped from a particular height, the ratio of distance covered by the body in successive seconds is 1 : 4 : 9.....

Related Questions : (Answer in comment box.)

Q1. Velocity at the top of vertical journey under gravity when a body is projected upwards with velocity 1000 m/s is 

(a) zero 

(b) 10 m/s 

(c) 100 m/s 

(d) 1000 m/s 

Q2. At a particular height , the velocity of an ascending body is u . The velocity at the same height while the body falls under gravity is 

(a)  u 

(b) 2u 

(c) u/2 

(d) u²

Q3. The acceleration of body in motion can be known from slop of

(a) force-time graph  

(b) work-time graph 

(c) displacement-time graph

(d) velocity-time graph

Q4. Two different masses are dropped from same height for downward journey under gravity . The larger mass reaches the ground in time t. The smaller mass takes time 

(a) equal to √t 

(b) greater than t

(c) less than t

(d) equal to t

Q5. The dot product of two vectors is equal to zero when they are inclined to each other at an angle of 

(a) 180° 

(b) 90° 

(c) 45°

(d) zero 

Q6. The cross product of two vectors gives zero when the vectors enclose an angle to 

(a) 90° 

(b) 180° 

(c) 45° 

(d) 120° 

Q7. Two vectors under scalar product yield one if angle between them is of  

(a) 90° 

(b) zero 

(c) 45 

(d) 60° 

Q8. Two vectors under vector multiplication have magnitude one , if angle between them is of 

(a) 45° 

(b) 90° 

(c) 60°

(d) 120°

Q9. Two vectors of magnitudes 10 and 15 units can never have a resultant equal to 

(a) 15 units 

(b) 3 units 

(c) 10 units 

(d) 20 units

Q10. The sum of two vectors A and B is at right angles to their difference . This is possible if 

(a) A = 2B

(b) A = B 

(c) A = 3B 

(d) B = 2A

| Vector and Motion |

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