More on Vectors, Rectilinear and Projectile Motion - with Figures, Examples, Solved Problems and MCQ Quizzes




Vectors, Rectilinear and Projectile motion (Also check out the MCQ Tests at the end of this page)

Target Audience: High School Students, College Freshmen and Sophomores, 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!

This might also be helpful in studying topics required by Common Core Physics. 


A Quick Summary of what we'll study in this chapter 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.)


Introduction
Often we need to work with quantities which have both numerical and directional properties. For example, in order to pilot a small plane, you need to know the speed of the wind as well as its direction. In order to go to a store to buy a pen, you need to know its distance from your current location as well the directions in which to proceed. Thus, directions play a significant role in various everyday phenomena. Vectors are just another mathematical tool for simplifying our understanding of such phenomena and they make our calculations a lot easier. Examples of vector quantities are displacement, velocity, force, momentum, etc. By contrast a scalar is a quantity that has magnitude but no direction. e.g; temperature of a body or its mass. Now vectors have their own algebra i.e. their own addition, subtraction and multiplication rules. There can be a cause of confusion here when trying to understand vector multiplication. Vectors multiply in two ways: dot product and cross product. You might think how can two physical quantities multiply in two different ways which produce totally different results. The important thing to remember here is that the result of these two different multiplications are again two different physical quantities and not one which we tend to assume out of habit while dealing with scalars.
Also, you must always keep in mind that vectors (and all the mathematics for that matter) and its rules came after observing the behavior of physical phenomenon in a particular fashion and not otherwise though we study them in reverse order.

 

 

 
Applet to demonstrate the general trajectory of a projectile on flat as well as inclined surfaces; with and without air resistance; starting with different velocities and at different angles.
 
 

Here are the topics covered in this tutorial :


Multiplication of a vector with a scalar

A vector A multiplied by a real number λ is also a vector, whose magnitude is λ times the magnitude of the vector A and whose direction is the same or opposite depending upon whether λ is positive or negative.

Addition of Vectors

To add two vectors (say A and B), they are drawn with their lengths proportional to their magnitudes and then one of the vectors (say B) is displaced parallel to itself so that its tail coincides with the head of the other vector (i.e. A). The tail of vector A is then joined to the head of B which is the resultant vector i.e. vector A+B.

Resolving a Vector into components : 

How to compute the projections of a vector along the axes We will see how Ax = ACosθAy = ASinθ. It is very easy to see that given the components, magnitude of a vector can be obtained using the relation,  A = √(Ax2 + Ay2) and θ = tan-1 (Ay/Ax). So, in all the problems on vectors, a vector will be specified either with its components Ax and Ay or its magnitude and direction i.e. A and θ.

Unit Vectors :

A unit vector is a dimensionless vector having a magnitude of exactly 1. Its purpose is to specify a given direction and it has no other physical significance. Symbols i, j, and k are used to represent unit vectors pointing in the positive x, y, and z directions, respectively.

Multiplication of Vectors : 

The scalar product (dot product) and the cross product (which is a vector)

Dot Product: Geometrically, the dot product of two vectors is the magnitude of one times the projection of the other along the first. The symbol used to represent this operation is a small dot at middle height (·), which is where the name "dot product" comes from. Since this product has magnitude only, it is also known as the scalar product.
A · B = AB Cosθ
Cross ProductGeometrically, the cross product of two vectors is the area of the parallelogram between them. The symbol used to represent this operation is a large diagonal cross (×), which is where the name "cross product" comes from. Since this product has magnitude and direction, it is also known as the vector product.
  A × B = AB Sinθ n̂  

Rectilinear Motion : 

We will cover motion without acceleration, motion with constant acceleration, motion with non-uniform acceleration

Projectile Motion : 

The parabolic trajectory of a projectile, projectile motion on inclined planes. An object that is in flight after being thrown is called a projectile. For example, if you throw a ball up in air at an angle other than 0o with the vertical, it follows a curved path which is called its trajectory. It becomes easier to analyze the path of a projectile if we make the following two assumptions: (1)the free-fall acceleration g is constant over the range of motion and is directed downward, and (2) the effect of air resistance is negligible. With these assumptions, we find that the trajectory of a projectile is always a parabola.



Here are some of the questions solved in this tutorial :


Q: A pedestrian moves 5.00 km east and then 12.0 km north. Using the graphical method, find the magnitude and direction of the resultant displacement vector.  

Q: Prove that  A.(AXB) = 0

Q: If A, B, C are mutually perpendicular, show that CX(AXB) = 0. Check if the converse is true.

Q: A ball is dropped from a balloon going up at a speed of 10 m/s when the balloon is at a height of 50 m/s. How long will the ball take to reach the ground? What will be the velocity of the ball when it strikes the ground? Take g = 10 ms-2.

Q: A wooden boxcar is moving along a straight railroad track at speed v1. A sniper fires a bullet (initial speed v2) at it from a highpowered rifle. The bullet passes through both lengthwise walls of the car, its entrance and exit holes being exactly opposite each other as viewed from within the car. From what direction, relative to the track, is the bullet fired? Assume that the bullet is not deflected upon entering the car, but that its speed decreases by 25%. Take v1 = 95 km/h and v2 = 700 m/s.

Q: A swimmer wishes to cross a river 550 m wide flowing at 6 km/h. His speed w.r.t. the water is 3.4 km/h. If he heads in a direction making an angle θ with the flow of the river, find the time taken. Also find the shortest time taken to cross the river.

Q: Ship A is located 4.0 km north and 2.5 km east of ship B. Ship A has a velocity of 20 km/h toward the south, and ship B has a velocity of 30 km/h in a direction 37° north of east. (a) What is the velocity of A relative to B in unit-vector notation? (b) Write an expression for the position of A relative to B as a function of t, where t = 0 when the ships are in the positions described above. (c) At what time is the separation between the ships least? (d) What is that least separation?

Q: In 1936 Olympic Games at Berlin (where g 9.8128 m/s2), a world’s running broad jump record of 8.09 m was established. Assuming the same values of v and θ, by how much would his record have differed if the same person had competed instead in 1956 at Melbourne (where g 9.7999 m/s2)?

Q: A ball is dropped from a height. It takes 0.15 s to cross the last 5 m before hitting the ground. Find the height from which it was dropped. Take g = 10 ms-2.

Q: A ball is thrown horizontally from a height of h = 30 m and hits the ground with a speed that is three times its initial speed. What is the initial speed? Take g = 10 ms-2.

Q: A person sitting on the top of a tall building is dropping marbles at linearly increasing intervals of 1s, 2s, 3s and so on. Find the position of the 4th, 5th and 6th marble when 7th marble is being dropped. Take g = 10 ms-2.

Q: A soccer ball is kicked from the ground with an initial speed of 25 m/s at an upward angle of 60°. A player 70 m away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground? Take g = 10 ms-2.

Q: A ball rolls horizontally off the top of a stairway with a speed of 1.8 m/s. The steps are 22 cm high and 22 cm wide. Which step does the ball hit first? Take g = 10 ms-2.

Q: An elevator is descending with uniform acceleration. A person in the elevator drops a marble at the moment the elevator starts to measure the acceleration of the elevator. The marble is 2 m above the floor when it is dropped. It takes 1.2 s to reach the floor of the elevator. What is the acceleration of the floor. Take g = 10 ms-2.

Q: A projectile is launched with a velocity 30 m/s at an angle θ above the plane which is inclined at an angle of 30o with the horizontal. For what value of ‘θ’ does it bounce up the plane. Assume elastic impact between the plane and the projectile. Take g = 10 ms-2.

Q: A projectile is launched with a velocity 30 m/s at an angle θ above the plane which is inclined at an angle of 30o with the horizontal. For what value of ‘θ’ does it bounce up the plane. Assume elastic impact between the plane and the projectile. Take g = 10 ms-2.

Q: Particle A moves along the line y = 30 m with a constant velocity of magnitude 5 m/s and parallel to the x axis. At the instant particle A passes the y axis, particle B leaves the origin with a zero initial speed and a constant acceleration of magnitude a = 0.50 m/s2. What angle θ between ‘a’ and the positive direction of the y axis would result in a collision?

Q: A bird is sitting on top of a lamp post. A stone is thrown so as to hit the bird from a distance of 2m from the lamp post at an angle of 30o with the horizontal. But the bird starts flying in the same direction with a velocity of 2 m/s as soon as the stone is launched. What was the initial velocity of the stone if it still hits the bird? Take g = 10 ms-2.

Complete Tutorial Document with Solved Problems (Also check out the MCQ Quizzes at the end of this page):



MCQ Quiz #1- Rectilinear Motion: Challenging Problems

Companion MCQ Quiz- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz #1- Rectinlinear Motion



MCQ Quiz #2- Projectile Motion- Challenging Problems

Companion MCQ Quiz- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz #2