Distance and Displacement - Concepts and Solved Examples for Class 11 Physics, JEE, NEET exams
Distance and Displacement
Welcome to our page on position, distance, displacement, and frames of reference in Physics for Class 11, JEE, and NEET exams. In this page, we will explore the meaning of position, distance, and displacement, as well as their importance in understanding frames of reference. You will also find solved examples to help you master these concepts. Let's dive in!
In physics, position refers to the location of an object relative to a reference point.
To locate the position in motion or at rest,we need a frame of reference.
Simplest way to choose a frame of reference is to choose three mutually perpendicular axis labelled as X-,Y- and Z- axis as shown in figure below
Such system of labelling position of an object is known as rectangular coordinates system
If A(x,y,z) be the position of any point in rectangular co-ordinates system it can be labelled as follows
Point O is the point of intersection of these mutually perpendicular axis and is known as reference point or origin
of frame of reference
To measure a time ,we can also attach a clock with this frame of reference
If any or all co-ordinates of the object under consideration changes with time in this frame of reference then the
object is said to be in a motion w.r.t the frame of the reference otherwise it is at rest
For describing motion in one dimension we need one set of co-ordinates axis i.e only one of X,Y and Z axis
Motion of an object along a straight line is an example of motion in one dimension
For such a motion, any one axis say X-axis may be choose so as to coincide with the path along which object is moving
Position of the object can be measured w.r.t origin O shown in the figure
Position to the right of the origin has positive values and those to the left of origin O has negative values.
Frame of reference
We know that we need a frame of reference is needed to locate the position of any object. For that we choose rectangular co-ordinate
system
of three mutually perpendicular axis that are $x$, $y$ and $z$ axis. The point of intersection of these three axes is called
Origin 'O'
and is considered as the Reference Point.
The $x$, $y$ and $z$ co-ordinates describe the position of object with respect to the co-ordinate system.
To measure time , we need a clock.
The co-ordinate system along with a clock constitutes a Frame of reference.
So , the frame of reference is a co-ordinate
system , with a clock w.r.t. which, an observer can describe the position, displacement etc. of an object.
Frames of reference are of two types:-
Inertial frame of reference:- These are the frame of reference in which Newton's First law of Motion is applicable.
Non-Inertial frame of reference:- These are the frame of reference in which , Newton's first law of motion is not applicable.
Distance and displacement:
In the graph shown below an object is at position P at time t_{1} and at position R at time t_{2}.
In the time interval from t_{1} to t_{2} particle has travelled path PQR and length of the path PQR is the
distance travelled by the object in the time interval t_{1} to t_{2}
Now connect the initial position of the object P with its final position R through a straight line and we get the displacement
of the object.
Displacement of the object has both magnitude and direction i.e., displacement is a vector quantity.
Magnitude of displacement vector is equal to the length of straight line joining initial and final position and its direction
points from the initial position of object towards its final position.
In contrast to displacement, distance is scalar quantity.
$\text {Distance} \ge \text{magnitude of Displacement}$
Difference between distance and displacement
For a moving particle in a given time interval distance can be many valued function, but displacement
would always be single valued function
Displacement could be positive, negative or zero, but distance would always be positive.
Displacement can decrease with time, but distance can never decrease with time.
Distance is always greater than or equal to the magnitude of displacement.
Distance would be equal to displacement if and only is particle is moving along straight line without any change in direction.
Solved examples on distance and displacement - Class 11 physics
Question 1
A object moves 8km in North Direction and then 6km in West direction
Find the distance and displacement of the object Solution
Distance=8 km + 6 km =14 km
Displacement is the shortest distance between the two points
From Pythagorous theorem
$D=\sqrt { 8^2 + 6^2}=10$
Question 2
A particle moves from point A to B along the circular path of radius R
Find the distance and displacement of the object Solution
Distance=$\pi R + \frac {\pi R}{2}= \frac {3\pi R}{2}$
Displacement is the shortest distance between the two points
From Pythagorous theorem
$D=\sqrt { R^2 + R^2}=R \sqrt 2$
Distance and Displacement solved examples for JEE and NEET
Given below are some of the questions on this topic of position, distance and displacement.
Qiestion 1 A particle starts from the origin, goes along the X−axis to the point (20 m,0) and then returns along the same line to the point (−20 m,0). Find the distance and displacement of the particle during the trip. Solution
Let $O$ be the starting point and coordinates are, $O(0, 0)$, $A(20, 0)$ and $B(-20,0)$
Distance travelled $= OA + AB = 20m + 40 m = 60 m$
Displacement = distance between final and initial point
$= OB = 20 m$ (negative direction)
Question 2 A man a has to go 50 m due north, 40 m due east and 20 m due south to
reach a field.
(a) What distance he has to walk to reach the field?
(b) What is his displacement from his house to the field? Solution
It is given in the question that
Distance moved in North $AB=50m$
Distance moved in East $BC=40m$
Distance moved in south $CD=20m$
(a) From the figure given above
Distance that man has to walk to reach the field $= AB + BC + CD = 50 + 40 + 20 = 110 $
So, man must walk $110 m$ to reach the field.
(b) Displacement from man's house to the field, which is AD
$AD = \sqrt{AE^2 + ED^2} = 50 m$
From the triangle AED
$\tan \phi = \frac{DE}{AE} = \frac{30}{40} = \frac{3}{4}$
or,
$\phi = \tan^{-1}(\frac{3}{4})$
Therefore, his displacement from his house to the field is 50 m, $\tan^{-1}(\frac{3}{4})$ north to east.
Distance and displacement assertion reason type questions
Assertion reason questions are now asked in various exams be it school class 11 exam or NEET entrance exam or JEE exam. Given below are few assertion reason based questions on this topic of position, distance and displacement.
Directions:
In each of the following questions, a statement of Assertion(A) is given and a corresponding statement of Reason(R) is given just below it. Of the statements, given below, mark the correct answer as: Options: A. Both A and R are true, and R is the correct explanation of A B. Both A and R are true, but R is not the correct explanation of A C. A is true, but R is false D. A is false, but R is true
Question 1:
Assertion (A): Distance is a scalar quantity.
Reason (R): Distance does not depend on the direction.
Question 2:
Assertion (A): Displacement can be zero even if the distance traveled is not zero.
Reason (R): Displacement depends on the initial and final position of the object, while distance depends on the path traveled.
Question 3:
Assertion (A): The magnitude of displacement can be greater than the distance traveled.
Reason (R): Distance is a scalar quantity, while displacement is a vector quantity.
Question 4:
Assertion (A): The position of an object is always specified relative to a reference point.
Reason (R): The reference point is necessary to define the position of an object in a coordinate system.
Question 5:
Assertion (A): The path length is always equal to the magnitude of the displacement.
Reason (R): Both path length and displacement describe the distance between two points.
Question 6:
Assertion (A): An object moving in a straight line has the same distance and displacement.
Reason (R): Displacement is the shortest distance between the initial and final position of the object.
Answers and Explainations
Answer 1: A. Both A and R are true, and R is the correct explanation of A Explanation: Distance is a scalar quantity as it only depends on the magnitude and does not consider the direction, while displacement is a vector quantity that takes direction into account.
Answer 2: A. Both A and R are true, and R is the correct explanation of A Explanation: Displacement is the shortest distance between the initial and final positions of an object. If an object returns to its starting position, the displacement will be zero, even if it has traveled a certain distance along a path.
Answer 3: D. A is false, but R is true Explanation: The magnitude of displacement can never be greater than the distance traveled. Displacement is the shortest distance between the initial and final positions, while distance is the total path traveled.
Answer 4: A. Both A and R are true, and R is the correct explanation of A Explanation: To describe the position of an object, a reference point is required. This reference point is typically the origin of a coordinate system, and the position of the object is given as coordinates relative to this point.
Answer 5: D. A is false, but R is true Explanation: The path length and magnitude of displacement are not always equal. Path length is the total distance traveled along the path by an object, taking into account all the turns and curves in the path. On the other hand, displacement is a vector quantity that measures the shortest distance between the initial and final positions of the object, considering both magnitude and direction.
In some cases, such as when an object moves in a straight line without changing direction, the path length and magnitude of displacement will be equal. However, in other cases where the object's path isn't straight or it changes direction, the path length will be longer than the magnitude of displacement. The reason statement is true because both path length and displacement do describe the distance between two points, but their definitions and implications are different, as mentioned earlier.
Answer 6: B. Both A and R are true, but R is not the correct explanation of A Explanation: When an object moves in a straight line, its distance and displacement are indeed the same. However, the reason statement is not the correct explanation for the assertion, as the reason statement explains the general definition of displacement.
Summary
Here is the summary of Distance and Displacement in Physics
Distance is the path length traversed by an object between two points
Displacement does not depend on path length and is the shortest distance between the two points
$\text {Distance} \ge \text{magnitude of Displacement}$