Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Physics B, AP Physics C, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!
Much of the content is also relevant to the needs of the Common Core standards in Physics, relevant to American students.
A Quick Summary of what we'll study in Vectors and the kind of problems we'll solve
(after this intro, there is a comprehensive document with study material as well as solutions to problems.)
Vectors are quantities which have both magnitude and direction. A vector can be 2 dimensional, 3 dimensional or more. Every vector has 2 parts : Magnitude and Direction
There are 2 ways to represent a vector : Graphical and Analytical
Graphical Representation of a Vector
A vector is represented by an arrow with the direction of arrow representing the direction of vector and length of arrow representing the magnitude.
Fig 1
Graphical representation of a vector
Analytical Structure of a Vector
This way of representing vectors is used while doing the analytical analysis of a problem. In this, the vectors are represented with respect to a coordinate system.
There are mainly 2 coordinate systems used:
- Cartesian coordinate system
- Polar coordinate system
Cartesian coordinate system
In this system the vector is represented by its components in various directions. For a 2D vector this would mean:
A = Ax + Ay & |A| =( (Ax)2 + (Ay)2 )(1/2) (bold represent vector)
fig 4.
Polar coordinate system
In the polar representation of vectors, r represents the magnitude of vector and ϴ denotes the direction of the vector with respect to the fixed coordinate system.
Polar coordinates could be converted to cartesian coordinates by simply taking components along the axes such as,
|Ax| = r sin ϴ & |Ay| = r cos ϴ and tan ϴ = AxAy
Unit Vector
A unit vector is a vector with magnitude 1. fig 5.
Vector Addition
.
fig 6
For vector addition of A and B, draw a vector from ending tip of A to starting tip of B. The resulting vector is the sum of the given vectors.
Analytically if,
A = Ax + Ay
B = Bx + By
A + B = (Ax + Bx) + (Ay + By)
Vector subtraction
Vector subtraction is similar to vector addition. For A - B just reverse the direction of B and then add A to it.
Resolution of vectors along coordinate axes
Let i ,j & k be unit vectors along the coordinate axes X, Y & Z. Then any vector could be resolved along these axes using these unit vectors. Let Ax ,Ay and Az be the components of A along X, Y and Z axes, then:
A = Ax i + Ay j + Az k
Component along another vector
Let b be a unit vector along B, then component of A along B is given by
component = |A.b|
Vector Dot Product
A.B = |A||B| cos ϴ = AxBx + AyBy + AzBz
Properties of vector dot product
1) A.A = |A|2
2) A.B = B.A
3) A.(B + C) = A.B + A.C
4) m(A.B) = (mA).B = A.(mB) = (A.B)m
Vector Cross Product
fig 9.
If C = AxB, then |C| = |A||B| sinϴ
Direction of C is perpendicular to both A & B ie. perpendicular to plane containing both A & B.
Angle between 2 vectors
Angle between 2 given vectors A and B = ϴ = cos-1(A . B|A||B|)
Rectilinear Motion
Rectilinear or planar motion could be thought of as vector sum of 2 linear motions. For simplicity,often any motion in a plane is resolved along the 2 coordinate. Both motions are then solved separately and finally added vectorially to get the final solution.
Concepts
Let position vector of a particle be r = x i + y j + z k
Then , velocity vector, v = drdt= (dxdt)i + (dydt)j + (dzdt)k
acceleration vector,a = dvdt = d2rdt2 = (d2xdt2)i + (d2ydt2)j + (d2zdt2)k
Then , velocity vector, v = drdt= (dxdt)i + (dydt)j + (dzdt)k
acceleration vector,a = dvdt = d2rdt2 = (d2xdt2)i + (d2ydt2)j + (d2zdt2)k
Relative velocity
Relative velocity of A with respect to B = vAB = vA - vB
Relative acceleration of A with respect to B = aAB = aA - aB
Constant acceleration motion
For a motion with constant acceleration, following relations are useful, derived from the differential equations mentioned above:
v = u + at here, u = initial velocity of particle, v = final velocity of particle
s = ut + 0.5at2 a = acceleration of the particle, s = displacement of the particle
v2 = u2 + 2as t = time passed since u
Projectile Motion
fig10.
Equation of projectile motion’s path where, u = magnitude of initial velocity of projectile
ϴ = angle made by initial velocity and X-axis
v = velocity of particle at any instant
t = time passed
x = displacement along x axis
y = displacement along y axis
g = acceleration due to gravity downward
Equation = y = x(tan ) - (g2 u2 cos2 ) x2
by equation, we could say that path of motion of a projectile is parabolic.
Formulas for projectile motion
Horizontal range of motion, R = u2 sin 2g (As can be seen, range is maximum for ϴ = 450 )
Maximum height , H = (u sin)22g
time of maximum height, t = u sing
time of flight, T = 2t = 2u singAngle between 2 given vectors A and B = ϴ = cos-1(A . B|A||B|)
Angle between 2 vectors
Angle between 2 given vectors A and B = ϴ = cos-1(A . B|A||B|)
Properties of vector dot product
1) A.A = |A|2
2) A.B = B.A
3) A.(B + C) = A.B + A.C
4) m(A.B) = (mA).B = A.(mB) = (A.B)m
Vector Cross Product
fig 9.
If C = AxB, then |C| = |A||B| sinϴ
Direction of C is perpendicular to both A & B ie. perpendicular to plane containing both A & B.
Direction of C is perpendicular to both A & B ie. perpendicular to plane containing both A & B.
Angle between 2 vectors
Angle between 2 given vectors A and B = ϴ = cos-1(A . B|A||B|)
Polar coordinates could be converted to cartesian coordinates by simply taking components along the axes such as,
|Ax| = r sin ϴ & |Ay| = r cos ϴ and tan ϴ = AxAy
Applying the Principles - Related Problems Set with Solutions at the End
1. Problem: Find angle between 2 vectors, A = i + j and B = -i + j ?
2. Problem: If A = 8 i + 4 j and B = 4 i + 1 j , find magnitude of A + B and A - B ?
3. Problem: Which one of the following statements is true?
(a) A scalar quantity is the one that is conserved in a process.
(b) A scalar quantity is the one that can never take negative values.
(c) A scalar quantity is the one that does not vary from one point to another in space.
(d) A scalar quantity has the same value for observers with different orientations of the axes.
4. Problem: In a river with current velocity 5 kmph in 600 east to south, a steamer is racing towards north with velocity 20 kmph. Find resultant velocity of boat.
5. Problem: The component of a vector r along X-axis will have maximum value if
(a) r is along positive Y-axis
(b) r is along positive X-axis
(c) r makes an angle of 45° with the X-axis
(d) r is along negative Y-axis
6. Problem: In a two dimensional motion, instantaneous speed V0 is a positive constant. Then which of the following are necessarily true?
(a) The average velocity is not zero at any time.
(b) Average acceleration must always vanish.
(c) Displacements in equal time intervals are equal.
(d) Equal path lengths are traversed in equal intervals.
7. Problem: If a stone is thrown with initial speed of 20 ms-1 at an angle of 450 with horizontal, find the horizontal range?(assume g = 10 ms-2 )
8. Problem: For the same stone as in previous question, find the maximum height achieved by the stone?
9. Problem: For problem 7,find the time of flight of the stone?
10. Problem: If in the problem 7, angle of throw is changed from 450 to 300 ,find the new horizontal range of throw?
11. Problem: The horizontal range of a projectile fired at an angle of 30° is 50 m.
If it is fired with the same speed at an angle of 45°, its range will be ?
12. Problem: A particle starts from the origin at t = 0 s with a velocity of 10.0 j ms-1 and moves in the x-y plane with a constant acceleration of ( 8.0 i + 4.0 j) m s-2 .At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
13. Problem: i and j are unit vectors along x and y- axis respectively. What are the components of a vector A= 2 i +3 j along the directions of i + j and i − j ?
14. Problem: Rain is falling vertically with a speed of 20 ms-1. Winds blowing after some time with a speed-1 of 20 ms-1 in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella ?
15. Problem: A hiker stands on the edge of a cliff 500 m above the ground and throws a stone horizontally with an initial speed of 20 ms-1. Neglecting air resistance, find the time taken by the stone to reach the ground, and the speed with which it hits the ground. (Take g = 10 ms-2 ).
16. Problem: A football is kicked into the air vertically upwards. What is its (a) acceleration, and (b) velocity at the highest point?
17. Problem: A fighter plane is flying horizontally at an altitude of 1 km with speed 360 km/h. At what angle of sight (w.r.t. horizontal) when the target is seen, should the pilot drop the bomb in order to attack the target?
18. Problem: A projectile is fired in such a way that its horizontal range is equal to twice its maximum height. What is the angle of projection?
19. Problem: The ceiling of a long hall is 20 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 ms-1 can go without hitting the ceiling of the hall ?
20. Problem:A cricketer can throw a ball to a maximum horizontal distance of 90 m. How much
high above the ground can the cricketer throw the same ball ?
Problem. 21: An athlete can throw the ball with a speed v. If he throws the ball while
running with speed u at an angle ϴ to the horizontal, find
(a)The effective angle to the horizontal at which the ball is projected in air as seen by a spectator.
(b)Time of flight?
(c)What is the distance (horizontal range) from the point of projection at which the
ball will land?
(d) Find ϴ at which he should throw the ball that would maximise the horizontal
range as found in (c).
Problem. 22: A fighter plane flying horizontally at an altitude of 1 km with speed 1080 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 ms-1 to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit ? (Take g = 10 m s-2 ).
