An object has moved through a distance. Can it have zero displacement? If yes, support your answer with an example.
Yes, an object can have zero displacement even after moving through a non-zero distance.
Key distinction:
• Distance is the total length of the path covered by the object. It is a scalar quantity and is always positive.
• Displacement is the shortest straight-line distance between the initial and final positions of the object, along with direction. It is a vector quantity and can be zero, positive, or negative.
Example: A person walks 5 m due east and then turns and walks 5 m due west, returning to the starting point.
• Total distance covered = 5 + 5 = 10 m
• Displacement = 0 m (initial and final positions are the same)
Similarly, an athlete completing one full lap of a circular track has covered a distance equal to the circumference of the track, but displacement = 0 because the starting and ending points coincide.
A farmer moves along the boundary of a square field of side 10 m in 40 s. What will be the magnitude of displacement of the farmer at the end of 2 minutes 20 seconds from his initial position?
Given: Side of square field = 10 m. Perimeter = 4 × 10 = 40 m. Time for one complete round = 40 s. Total time = 2 min 20 s = 140 s.
Number of rounds completed: 140 ÷ 40 = 3.5 rounds
After 3 complete rounds, the farmer is back at the starting corner (A). In the remaining 0.5 round (half a round = 20 m = two sides), the farmer travels along two sides of the square and reaches the diagonally opposite corner (C).
Displacement = diagonal of the square:
The magnitude of displacement = 10√2 ≈ 14.14 m (directed diagonally from corner A to corner C).
Which of the following is true for displacement? (a) It cannot be zero. (b) Its magnitude is greater than the distance travelled by the object.
Both statements are false.
(a) "Displacement cannot be zero" — FALSE:
Displacement can be zero. Whenever an object returns to its starting point (initial position = final position), the displacement is zero, even though the distance covered may be substantial. Example: One full round of a circular track.
(b) "Magnitude of displacement is greater than distance" — FALSE:
The magnitude of displacement is always less than or equal to the distance covered (|displacement| ≤ distance). It is equal to the distance only when the object moves in a straight line in a single direction without turning back. Displacement can never exceed distance because the straight-line path (displacement) is always the shortest path between two points.
Distinguish between speed and velocity.
Speed:
(i) Scalar quantity — has only magnitude, no direction.
(ii) Defined as the total distance covered per unit time: Speed = distance ÷ time.
(iii) Always positive (or zero); can never be negative.
(iv) SI unit: m/s (metres per second).
(v) Example: A car moving at 60 km/h has a speed of 60 km/h.
Velocity:
(i) Vector quantity — has both magnitude and direction.
(ii) Defined as the displacement per unit time: Velocity = displacement ÷ time.
(iii) Can be positive, negative, or zero depending on direction of motion.
(iv) SI unit: m/s (metres per second).
(v) Example: A car moving at 60 km/h towards the north has a velocity of 60 km/h north.
Key: two objects can have the same speed but different velocities if they are moving in different directions.
Under what condition(s) is the magnitude of average velocity of an object equal to its average speed?
The magnitude of average velocity equals average speed when the object moves in a straight line in one direction without reversing.
In this condition, the total path length (distance) equals the magnitude of displacement, so:
Explanation: Average speed = total distance / total time. Average velocity = total displacement / total time. These are equal only when distance = |displacement|, which happens when the object moves in a straight line without changing direction. If the object turns or reverses, the distance becomes greater than |displacement|, making average speed > |average velocity|.
What does the odometer of an automobile measure?
The odometer of an automobile measures the total distance travelled by the vehicle (not displacement). It records and displays the cumulative length of the path covered since the vehicle was manufactured or since the odometer was last reset.
Since distance is the total path length regardless of direction, the odometer keeps adding up the distance whether the car goes straight, turns, reverses, or takes a curved path. The odometer is typically measured in kilometres (km) or miles.
The speedometer, on the other hand, measures the instantaneous speed (not velocity) of the vehicle at any given moment.
What does the path of an object look like when it is in uniform motion?
An object in uniform motion moves with constant velocity — it covers equal distances in equal intervals of time and moves in a straight line (no change in direction).
Path: The path of the object is a straight line.
Distance-Time graph: A straight line with a constant positive slope. The slope equals the constant speed of the object. The steeper the line, the greater the speed.
Velocity-Time graph: A horizontal straight line (parallel to the time axis) at the value of the constant velocity, indicating that velocity does not change with time. The area under this graph gives the distance covered.
During an experiment, a signal from a spaceship reached the ground station in five minutes. What was the distance of the spaceship from the ground station? The signal travels at the speed of light, that is, 3 × 108 m/s.
Given: Speed of signal = 3 × 108 m/s. Time = 5 minutes = 5 × 60 = 300 s.
Using: Distance = Speed × Time
The spaceship was at a distance of 9 × 1010 m (90 billion metres or about 0.6 AU) from the ground station.
When will you say a body is in (i) uniform acceleration? (ii) non-uniform acceleration?
(i) Uniform Acceleration:
A body is said to be in uniform acceleration when its velocity changes by equal amounts in equal intervals of time. The acceleration remains constant throughout the motion. This means the body speeds up (or slows down) at a steady rate.
Example: A ball falling freely under gravity (ignoring air resistance) accelerates uniformly at g = 9.8 m/s2. On a velocity-time graph, uniform acceleration appears as a straight line with a constant slope.
(ii) Non-uniform Acceleration:
A body is said to be in non-uniform acceleration when its velocity changes by unequal amounts in equal intervals of time. The acceleration varies (changes) with time.
Example: A car in city traffic that speeds up, slows down, and stops at varying rates. A ball rolling down an irregular inclined surface. On a velocity-time graph, non-uniform acceleration appears as a curved line (not a straight line).
A bus starting from rest moves with a uniform acceleration of 0.1 m/s2 for 2 minutes. Find (a) the speed acquired, (b) the distance travelled.
Given: Initial velocity u = 0 (starts from rest). Acceleration a = 0.1 m/s2. Time t = 2 min = 2 × 60 = 120 s.
(a) Speed acquired — using v = u + at:
(b) Distance travelled — using s = ut + ½at2:
The bus acquires a speed of 12 m/s and travels a distance of 720 m.
A train is travelling at a speed of 90 km/h. Brakes are applied so as to produce a uniform acceleration of −0.5 m/s2. Find how far the train will go before it is brought to rest.
Given: Initial speed u = 90 km/h = 90 × (1000/3600) = 25 m/s. Final velocity v = 0 (comes to rest). Acceleration a = −0.5 m/s2 (retardation).
Using v2 = u2 + 2as:
The train will travel 625 m before coming to rest.
A trolley, while going down an inclined plane, has an acceleration of 2 cm/s2. What will be its velocity 3 s after the start?
Given: Initial velocity u = 0 (starts from rest). Acceleration a = 2 cm/s2. Time t = 3 s.
Using v = u + at:
The velocity of the trolley after 3 s is 6 cm/s (down the incline).
A racing car has a uniform acceleration of 4 m/s2. What distance will it cover in 10 s after the start?
Given: Initial velocity u = 0 (starts from rest). Acceleration a = 4 m/s2. Time t = 10 s.
Using s = ut + ½at2:
The racing car covers a distance of 200 m in 10 s.
A stone is thrown in a vertically upward direction with a velocity of 5 m/s. If the acceleration of the stone during its motion is 10 m/s2 in the downward direction, what will be the height attained and how much time will it take to reach there?
Given: Initial velocity u = 5 m/s (upward). Final velocity v = 0 (at maximum height). Acceleration a = −10 m/s2 (downward, opposing upward motion).
Time to reach maximum height — using v = u + at:
Height attained — using v2 = u2 + 2as:
The stone reaches a maximum height of 1.25 m in a time of 0.5 s.
What is the nature of the distance-time graphs for uniform and non-uniform motion of an object?
Uniform Motion (constant speed):
The distance-time graph is a straight line (linear graph) with a constant positive slope. This is because in uniform motion, equal distances are covered in equal time intervals, so the ratio of distance to time (slope = speed) remains constant throughout. The steeper the line, the greater the speed. If the object is at rest, the graph is a horizontal straight line (zero slope).
Non-uniform Motion (changing speed):
The distance-time graph is a curved line (non-linear graph) because the distance covered per unit time is changing. The slope (which represents instantaneous speed) keeps changing:
• If the object is accelerating (speeding up), the curve bends upward (concave upward — increasing slope).
• If the object is decelerating (slowing down), the curve bends toward the horizontal (concave downward — decreasing slope).
The slope of the tangent at any point on the curve gives the instantaneous speed of the object at that moment.
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