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CLASS XI PHYSICS VOLUME - I NEET 40 2. MOTION IN A STRAIGHT LINE SESSION - 1 DISTANCE, DISPLACEMENT AND TYPES OF VELOCITY 1.1 DISTANCE, DISPLACEMENT AND TYPES OF VELOCITY DISTANCE: 1. Definition: Distance is the total path length covered by a particle during its motion, irrespective of the direction of motion. It measures "how much ground” an object has covered. 2. Key Characteristics: It is a scalar quantity, meaning it only has magnitude and no direction. Always a positive value or zero, as it represents the actual length of the path. Independent of the particle's direction of motion. 3. Symbol and SI Unit: Represented by the symbol S. Measured in meters (m) in the International System of Units (SI). Example: If a particle moves along a circular track with a radius of 5 m and completes one full revolution, the total distance traveled is equal to the circumference of the circle, which is 2π = 2π × 5 = 31.4 m. DISPLACEMENT Displacement is defined as the shortest straight-line distance between the initial and final positions of a particle, along with the direction from the starting point to the endpoint. It indicates the overall change in the position of the particle during its motion. 1. Key Characteristics: Displacement is a vector quantity, meaning it has both magnitude and direction. Its value can be positive, negative, or zero, depending on the relative position of the final point with respect to the initial point. It is always less than or equal to the distance traveled, as it represents the shortest path between two points. OA = r initial positionvector 1 OB = r final positionvector 2 AB r r; Displacement vector Hence displacement is “final position vector” minus “initial position vector”. Displacement can have zero magnitude, but distance travelled can never be zero. If r a i b j c k 1 1 1 1    and r a i b j c k 2 2 2 2    then r r r a i b j c k a i b j c k 2 1 2 2 2 1 2 1       r a a i b b j c c k 2 1 2 1 2 1    Example of Displacement: The shortest straight-line distance between the starting point (home) and the final point (home) is 0 m, because the person ends up at the same place where they started. Displacement = 0 m. Comparison between Distance and Displacement Aspect Distance Displacement Type Scalar (magni- tude only) Vector (magnitude and direction) Path De- pendence Depends on the actual path taken Depends only on the initial and final positions Value Always positive or zero Can be positive, negative, or zero Relation to Each Other Always greater than or equal to displacement Always less than or equal to distance Example Total path length covered Shortest straight-line distance between points SPEED AND VELOCITY Speed Speed is defined as the distance traveled by an object per unit time. It represents how fast an object is moving without considering the direction of motion. Speed is a MOTION IN A STRAIGHT LINE 2
CLASS XI PHYSICS VOLUME - I 41 NEET 2. MOTION IN A STRAIGHT LINE scalar quantity, which means it only has magnitude and no direction. Symbol: V Sl Unit: ms-1 Velocity Velocity is defined as the displacement of an object per unit time. It specifies both the rate of change of position and the direction of motion, making it a vector quantity. The direction of velocity is the same as the direction of displacement. Symbol: v  Sl Unit: ms-1 Let us consider an example to discuss their concepts: Let a particle move from point A to point B as shown: Along path 1, distance travelled is 100m, time taken is 10s Along path 2, distance is 200m, time taken is 15s Along path 3, distance travelled is 150m, time taken is 12s Along path 1: distance = s = 100m: Speed = 100/10ms-1 = 10 ms-1; Velocity = v m s f rom Ato B 10 1 (Displacement is same distance along path 1) Along path 2: displacement = 100 m; time = 15 s speed 200 15 40 3 1 ms ; Velocity v m s f rom Ato B 100 15 20 3 1 Along Path 3: Displacement = 100 m; distance = 150m; time = 12s Speed ms 150 12 75 6 1 ; velocity ms 100 12 25 3 1 (from A to B) Note: Regardless of the path taken (Path 1, Path 2, or Path 3), the direction of displacement and velocity remains constant, as they are determined by the straight-line vector from A to B. This highlights that while speed varies with the actual path, velocity is solely dependent on the displacement and its direction, reinforcing the distinction between these two fundamental concepts. AVERAGE SPEED Definition: Average speed is the total distance traveled by an object divided by the total time taken to cover that distance. It applies when an object moves with varying speeds over different time intervals or path segments. The formula is: Average Speed Total Total TimeTak DistanceCovered During Journey = entoComplete Journey Cases for Calculating Average Speed 1. When Distances and Speeds are mentioned: If an object covers distances s1 , s2 , ....., sN with corresponding speeds v1 , v2 ,....., vN, the formula for average speed is: Average Speed s s s s v s v s v N N N 1 2 1 1 2 2 ... ... 2. When Distance and Time intervals are mentioned: If an object travels distances s1 , s2 ,...., sN in respective time intervals t1 , t2 , ...., tN, the formula becomes: Average Speed s s s t t t N N 1 2 1 2 ... ... 3. When Speeds and Time intervals are mentioned: If the object moves at speeds v1 , v2 ,....., vN for respective time intervals t1 , t2 , ....., tN, the formula is: Average Speed v t v t v t t t t N N N 1 1 2 2 1 2 ... ... Special Cases of Average Speed 1. When Distances Are Equal: If the object travels equal distances (s1 = s2 = ••• = sN) at different speeds (v1 , v2 , • • •,vN), the formula for average speed simplifies to the harmonic mean of velocities: Average Speed N v v vN 1 1 1 1 2 ... Here, N is the total number of segments. 2. When Time Intervals Are Equal: If the object travels for equal time intervals (t1 = t2 = • • • = t N) at different speeds (v1 ,v2 ,..., vN), the formula for average speed becomes the arithmetic mean of velocities: Average Speed v v v N N 1 2 ... When Distances & Speeds are Mentioned When Distance & Time intervals are mentioned When Speed & Time intervals are mentioned Average speed v s s s s v s v s v N N N 1 2 1 1 2 2 ...... ...... Average Speed v s s s t t t N N 1 2 1 2 ..... ..... Average Speed v t v t v t t t t N N N 1 1 2 2 1 2 ....... ..... Average Velocity Definition: Average velocity is defined as the total displacement of an object divided by the total time taken. It represents the net change in position per unit time. AverageVelocity Displacement TimeTaken r t  Here, ∆r  is the displacement vector, and ∆t is the total time. Unlike average speed, average velocity is a vector quantity and includes direction. Key Observations When distances are constant across segments, the harmonic mean formula is used for calculating average speed. On the other hand, when time intervals are constant, the arithmetic mean formula is applied. Average speed focuses solely on the total path length and time taken, while average velocity considers the net displacement, making it inherently dependent on direction. These distinctions are crucial for understanding motion in physics and analyzing the nature of an object's movement under different conditions.
CLASS XI PHYSICS VOLUME - I NEET 42 2. MOTION IN A STRAIGHT LINE Instantaneous Velocity and Instantaneous Speed: Instantaneous velocity v  is a vector quantity that defines the rate of change of the position vector r  of a particle with respect to time. It provides both the magnitude and direction of the motion at a specific instant. The vector form of instantaneous velocity is expressed as: v r t d r dt t    lim 0 Here: ∆r  is the change in the position vector during the time interval ∆t d r  is the infinitesimal change in the position vector as ∆t approaches zero, r  Represents the position vector of the particle at any given time t. Instantaneous speed is the magnitude of the instantaneous velocity vector and is given by: v vvv x y z  222 Where vx , vy , vz are the components of the velocity vector along the x, y, and z-axes, respectively. Thus, instantaneous velocity not only describes how fast an object is moving but also specifies the direction of motion, while instantaneous speed represents the scalar magnitude of that motion. This vector formulation is critical for analyzing motion in multiple dimensions. Note: Always it is assumed that the particle is along a straight line path in this infinitesimal time. Uniform and Non-uniform Speed and Velocity Uniform Speed: A particle is said to have uniform speed if it covers equal distances in equal intervals of time, no matter how small the time intervals are. This implies that the rate of change of distance with time remains constant throughout the motion. Mathematically: VAB = VAC = VCB = VDE Uniform speed is independent of the path taken by the particle and can occur along straight lines or curved paths. EXAMPLE: A car moving at a constant speed of 60km/h on a straight road demonstrates uniform speed as it covers equal distances in equal time intervals. Non-uniform Speed A particle is said to have non-uniform speed if it covers unequal distances in equal intervals of time, irrespective of how small the intervals are. In this case, the rate of change of distance with time varies throughout the motion. Non-uniform speed is commonly observed in real-world scenarios where an object's motion is influenced by varying forces. Non-uniform Velocity A particle is said to have non-uniform velocity if it undergoes unequal displacements in equal intervals of time, regardless of the duration of these intervals. Unlike speed, velocity also depends on the direction of motion. Non- uniform velocity occurs when either the magnitude or the direction of velocity (or both) changes over time. Uniform velocity is possible only for motion along a straight line where both the magnitude and direction remain constant. In contrast, non-uniform velocity is common in curved paths or when an object accelerates or decelerates. EXAMPLE: A cyclist traveling uphill and downhill at varying speeds shows non-uniform speed because they cover unequal distances in equal time intervals. A train traveling at 80 km/h due east on a straight track exhibits uniform velocity as both its speed and direction remain constant. A car turning around a curve at a constant speed of 50 km/h or a ball thrown vertically upwards are examples of non-uniform velocity because either the direction or the magnitude of velocity changes with time. FORMULAS FOR AVERAGE SPEED Average speed is the total distance divided by total time. Vavg = Total distance travelled/Total time taken If a body travels a distance s1 in time t1 , s2 in time t2 and s3 in time t3 then the average speed is v s s s t t t avg 1 2 3 1 2 3 If an object travels distance s1 , s2 , s3 ect. With speeds v1 , v2 , v3 respectively in the same direction. Then, Average speed = s s s s v s v s v 1 2 3 1 1 2 2 3 3 + + + + If an object travels first half of the total journey with a speed and next half with a speed then its average speed is v s s s v s v x s v s v v v v v v v avg 1 2 1 2 1 2 1 2 1 2 2 2 1 1 2 If a body travels first 1/3rd of the distance with a speed v1 and second 1/3rd of the distance with a speed v2 and last 1/3rd of the distance with a speed v3 , then the average speed v sss s v s v s v avg 333 3 3 3 1 2 3 v v v v v v v v v v avg 3 1 2 3 1 2 2 3 3 1 If an object travels with speeds v1 ,v2 ,v3 etc., during time intervals t1 ,t2 ,t3 etc., then its average speed v t v t v t t t t 1 1 2 2 3 3 1 2 3 .... .... If t1 = t2 = t3 = ..... = t, then, v v t v t v t nt v v n avg 1 2 3 .... 1 2 ... i.e. The average speed is equal to the arithmetic mean of individual speeds. The actual path length traversed by a body is called distance. Note: v = vx i + vy j + vz k If varies with time t then Integrating on both sides then dr vdt v dr dt ∵ Integrating on both sides then dr vdt r r t t i f 1 2 ⇒ Displacement of the particle from time t1 to t2 is given by s r r V idt v jdt v kdt f i x y t t z t t t t 1 2 1 2 1 2
CLASS XI PHYSICS VOLUME - I 43 NEET 2. MOTION IN A STRAIGHT LINE 1.1 DISTANCE, DISPLACEMENT AND TYPES OF VELOCITY LEVEL - 0 1. The numerical ratio of displacement to distance is A) Always less than 1 B) Always greater than 1 C) Always equal to 1 D) May be less than 1 or equal to one 2. The location of a particle is changed. What can we say about the displacement and distance covered by the particle? A) Both cannot be zero B) One of the two may be zero C) Both must be zero D)Both must be equal 3. Consider the motion of the tip of the minute hand of a clock. In one hour A) the displacement is zero B) the distance covered is zero C) the average speed is zero D) the average velocity is zero A)a& b are correct B)a,b& c are correct C)a& d are correct D)b,c& d are correct 4. The numerical value of the ratio of average velocity to average speed is A)always less than one B) always equal to one C)always more than one D) equal to or less than one 5. If a particle moves in a circle describing equal angles in equal intervals of time, then the velocity vector A)remains constant B) changes in magnitude C)changes in direction D) changes both in magnitude and direction. LEVEL - 1 6. A Body moves 6m north. 8m east and 10m vertically upwards, what is its resultant displacement from initial position A) 10√2 m B) 10m C) 10 2 m D) 10 x 2m 7. A man goes 10m towards North, then 20m towards east then displacement is A) 22.5m B) 25m C) 25.5m D) 30m 8. A particle is constrained to move on a straight line path. It returns to the starting point after 10s. The total distance covered by the particle during this time is 30 m. Which of the following statements about the motion of the particle is false? A) Displacement of the particle is zero B) Average speed of the particle is 3 m/s C) Displacement of the particle is 30 m D) Both A and B 9. Which of the following options is correct for the object having a straight line motion represented by the following graph A) The object moves with constantly increasing velocity from O to A and then it moves with constant velocity B) Velocity of the object increases uniformly C) Average velocity is zero D) The graph shown is impossible 10. A particle moves for 20 s with velocity 3 m/s and then velocity 4 m/s for another 20 s and finally moves with velocity 5 m/s for next 20 s. What is the average velocity of the particle A) 3 m/s B) 4 m/s C) 5 m/s D) Zero LEVEL - 2 11. An ant starts from one corner of a cube of side length 3m and reaches the diagonally opposite corner. The displacement is? A) 8 3m B) 5 3m C) 3 3m D) 4 3m 12. A wheel of radius 1 meter rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is A) 2p B) 2π C) 2 4 D) p 13. If a cyclist takes one minute to complete half revolution on a circular path 120m radius, what is the average velocity? A) 1 m/s B) 2 m/s C) 3 m/s D) 4 m/s 14. A car covers the 1st half of the distance between two places at a speed of 40 km/hr and the second half at 60 km/hr. The average speed of the car is A) 100 km/hr B) 48 km/hr C) 50 km/hr D) 25 km/hr LEVEL-3 15. A particle constrained to move on a straight line path. It returns to the starting point after 10s. The total distance covered by the particle during this time is 30m. Which of the following statements about the motion of the particle is correct? a) displacement of the particle is zero b) average speed of the particle is 3 ms-1 c) displacement of the particle is 30m d) both a and b A) a is correct B) b is correct C) c is correct D) d is correct 16. Choose the correct statements from the following. A) The magnitude of instantaneous velocity of a particle is equal to its instantaneous speed B) The magnitude of the average velocity in an interval is equal to its average speed in that interval. C) It is possible to have a situation in which the speed of the particle is never zero but the average speed in an interval is zero. D) It is possible to have a situation in which the speed of particle is zero but the average speed is not zero. 17. A Particle located at x = 0 at time t = 0 , starts moving along with the positive x-direction with a velocity ‘v’ that varies as v x . The displacement of the particle varies with time as A) t2 B) t C) t1/2 D) t3 1.1 DISTANCE, DISPLACEMENT AND TYPES OF VELOCITY LEVEL - 1 1. An athlete completes one round of a circular track of radius R in 40 s. What will be his displacement at the end of 2 min. 20 s ? A) Zero B) 2R C) 2pR D) 7pR