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Velocity of an object is the rate at which an object's position varies with a frame of reference and time. The terms velocity and speed describe how quickly or slowly an object moves. Velocity of an object is responsible to indicate its position as well as how its position varies with time. It is often defined as the distance covered by an object in a unit time. Velocity can be simply by the formula: r = d/t
Where,
- r = rate (often also denoted as v, symbolising velocity)
- d = distance the object moved
- t = time the object took
For example, if a car travels towards the north at 7 meters per second (m/s), then its velocity will be equal to 7 m/s to the north. Clearly, velocity shows the position variation of an object with respect to time.
Read More: Unit of Acceleration
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Key Terms: Velocity, Speed, Acceleration, Displacement, Unit of Time, Uniform Velocity, Variable Velocity, Average Velocity, Instantaneous Velocity
What is Velocity?
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Velocity of an object is the rate at which an object's position varies with a frame of reference and time. Velocity is often expressed as the displacement of an object in a unit time. Velocity is a vector quantity, which means we need both magnitude (speed) and direction to define velocity. It has the SI unit of meter per second (ms-1). When the amount or direction of a body's velocity changes, the body is said to be accelerating.
Motion in a Straight Line Video Explanation
Formula of Velocity
The general formula of velocity is:
| v = d/t |
Where,
- v = Velocity (r or rate is also referred to in place of v)
- d = Distance covered by the object
- t = Time taken by the object to cover the distance
Speed and Velocity
Types of Velocity
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There are several types of Velocity, including:
- Uniform Velocity
- Variable Velocity
- Average Velocity
- Instantaneous Velocity
Uniform Velocity
If a particle covers equal displacements in similar intervals of time, it is said to be travelling with uniform velocity; that is, velocity is constant throughout the motion. That is, when a body moves at a constant speed, the magnitude and direction of velocity stay constant at all times.
In this situation, the product of the particle's velocity "v" and the time interval "t" yields the particle's displacement "s" over a given time interval "t."
Uniform Velocity
\(\therefore\) The formula of Uniform Velocity is ⇒ Displacement = Velocity \(\times\) Time Interval
Thus,
| s = vt |
Non-uniform Velocity or Variable Velocity
If the speed or direction of a body changes with time, it is said to be moving with variable velocity. Consider a body moving at a constant speed around a circle. Because its direction changes with motion, its velocity is changeable.
Variable Velocity
Average Velocity
The ratio of total displacement to the whole period is the average velocity.
Thus, \({{Total\ Displacement} \over Total\ Time \ Taken} = Average\ Velocity\) (vab)
As, Displacement = Average Velocity \(\times\) Time
Hence, s = vab \(\times\) t
In that instance, we can state that an object's average velocity over a particular time interval is the ratio of its displacement throughout that time interval to the time taken. Consider an item travelling on the X-axis. Let x(t1) and x(t2) be the coordinates of its position at times t1 and t2, respectively.
Displacement = x (t2) – x (t1)
⇒ Time Taken = t2 – t1
Hence, Average Velocity = x (t2) – x (t1)/(t2) – (t1)
Questions Related to Average Velocity and SpeedQues 1. In order to complete a journey of 200 km, a truck needs at least 4 hrs, thus calculate its average speed.
Ans. The correct answer is 50 km/hr. Explanation: Average speed = total distance/total time taken. Thus, total distance = 200 km Total time taken = 4 hrs Hence, the average speed = 200/4 = 50 km/hr. Ques 2. Assuming that a body falls under gravity with terminal velocity, what happens to the average velocity? Ans. Assuming that a body falls under gravity, the velocity does not vary when the body moves with terminal velocity. Meaning that in equal intervals of time, the average velocity remains constant. |
Instantaneous Velocity
When an object's instantaneous velocity is measured, it tells us how fast it moves at different places in a specific time interval. The magnitude of velocity is the speed at which something happens right now. At that point, the magnitude of the instantaneous velocity equals the instantaneous speed.
Instantaneous Velocity
| v(t) = d/dt (x (t)) |
Instantaneous velocity is a vector having a length per time dimension, just like average velocity.
Initial and Final Velocity
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Initial velocity of an object specifies how fast it travels when gravity first exerts a force on it. Final velocity, on the other hand, is a vector quantity that indicates a moving body's direction and speed after it has reached maximum acceleration.
How to Calculate the Initial Velocity?
The object's initial velocity (vi) is its velocity before it changes according to acceleration. As a result, the formula below can be used to calculate it.
| vi = vf – a \(\times\) t |
Where,
- vi = Initial Velocity (in m/s)
- vf = Final Velocity (in m/s)
- a = acceleration (in m/s2)
- t = time taken by the object (in s)
How to Calculate the Final Velocity?
In other words, the final speed of an object is equal to its starting velocity plus acceleration multiplied by the time it travelled. So,
| v = u + aΔt |
Where,
- v = final velocity
- u = initial velocity
- a = acceleration
- Δt = time difference
What is Constant Velocity?
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Motion with constant velocity can be considered the simplest form of motion. Constant velocity can be defined as:
| “The velocity which a particle attains because of a particle crossing an equal linear path at certain intervals of time.” |
Constant Velocity
An object moves with constant velocity if its speed has not changed even though the direction has. Simply, constant velocity can be expressed as a motion with zero acceleration. The graph shows displacement versus time for a body which moves with constant velocity. The straight line in the graph can be expressed as:
⇒ x = x0 + vt
As per the equation,
- x0 = displacement at time t,
- v = constant velocity of the body
Thus, \(\begin{array}{l}v=\frac{dx}{dt}\end{array}\)
Units of Velocity
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The SI unit for velocity is m/s or ms-1. Further information about velocity units and dimensions can be found in the table below.
| Units of Velocity | Values of Velocity |
|---|---|
| SI Unit | m/s |
| Other Units | ft/s, mph |
| Common Symbol | r, v, v |
| Dimensional Formula | [LT-1] |
Velocity and Speed
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For most of us, the concepts of speed and velocity are a little puzzling. Here, we explain how both concepts are different.
- The variation between speed and velocity is that speed informs us how rapidly an object is going, but velocity tells us not only how fast an object is moving but also in which direction it is travelling.
- Velocity is a function of displacement, whereas speed is a function of distance travelled.
- Velocity of a body at any particular time is known as instantaneous velocity.
- The average velocity is calculated by multiplying the total displacement by the entire time, v = xt, where x is the total displacement of the body and t is the time.
- Average speed is always less than or equal to average velocity; this is because, displacement can never exceed the distance travelled, yet the distance travelled can exceed displacement.
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Example of Velocity
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Let's look at an example to help you grasp the difference between instantaneous and average velocity.
Every morning, Shweta rides in her mother's car to school. Her school is 10 kilometres away, and it takes her 20 minutes to get there, but the speedometer on the dashboard of her automobile always shows a different value. How would she know her velocity?
So, here's how we figure out what Shweta's car's average speed is:
We've assumed the car is moving in a straight line for simplicity's sake, and we'll convert all time units to hours. As a result, 20 minutes equals 0.33 hours.
v = x \(\times\) t, average velocity
⇒ v = 10 km/0.33 hours
⇒ v = 30.30 km/hour
We can now see that, even if the car's speed varies, if it covers the same amount of distance in the same amount of time, its average velocity remains constant.
Solved Example Related to VelocityExample: Determine the average velocity considering the changes in displacement in three consecutive instances are 8 m, 10 m, 12 m (assume, total time taken = 6 s). Ans. As per the question, Total change in displacement, (8 + 10 + 12) = 30 m And, total time taken = 6 s Hence, Average velocity = total change in displacement/total time taken = 30/6 = 5 m/s. |
Difference between Velocity and Speed
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The following table provides a full comparison between velocity and speed.
| Velocity | Speed |
|---|---|
| Velocity of a body or an object determines its movement direction. | Speed refers to the quantitative measurement of how swiftly something moves. |
| Moving objects can have zero velocity. | A moving entity's speed can never be negative. |
| The displacement of an item in unit time is defined as velocity. | Speed is defined as the distance travelled by an object in a given amount of time. |
| It is the rate of displacement change. | It is the pace at which distance changes. |
| Velocity is a type of vector quantity. | Speed is a type of scalar quantity. |
Previous Year Questions
- The position x of a particle varies with time … [NEET 1997]
- A body starts from rest, what is the ratio … [NEET 1993]
- A boy standing at the top of a tower of … [NEET 2011]
- A bus is moving with a speed of … [NEET 2009]
- A car accelerates from rest at a constant rate … [NEET 1994]
- A car covers the first half of the distance between two places … [NEET 1990]
- A car is moving along a straight road with a uniform acceleration … [NEET 1988]
- A car moves a distance of 200 m. It covers the first half … [NEET 1991]
- A car moving with a speed of 40 km/h can be stopped … [NEET 1998]
- A man throws balls with the same speed vertically upwards … [NEET 2003]
- A particle moves along a straight line such that its displacement … [NEET 1994]
- A particle moves in a straight line with a constant acceleration … [NEET 1988]
- A particle moving along x-axis has acceleration … [NEET 2007]
- A particle of unit mass undergoes one dimensional motion … [NEET 2015]
- A stone falls freely under gravity. It covers distances … [NEET 2013]
Read Important PDF:
Velocity Important PDF
Things to Remember
- Velocity of a body is the rate at which an object's position changes with a frame of reference and time.
- Velocity is considered as a function of displacement, while speed is a function of distance that is travelled.
- To define velocity, we need both magnitude and direction.
- Instantaneous velocity is the speed at any given time, whereas average velocity is the total displacement divided by total time.
- Acceleration is indicated by a changing velocity.
- The velocity-time graph is used to show why an object accelerates at a constant rate.
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Sample Questions
Ques. Prove that the average velocity of a particle over an interval of time is either smaller than or equal to the average speed of the particle over the same interval. (2 Marks)
Ans. Average velocity is defined as the ratio of the total displacement to the total time. Average speed is defined as the ratio of the total distance to the total time. Since displacement is less than or equal to the distance, therefore the average velocity is less than or equal to the average speed.
Ques. A person being chased by a lion is running in a straight line towards his car at a constant speed of 4m/s. The car is at a distance of d meters away from the person. The lion is 26 m behind the person and running at a constant speed of 6 m/s. The person reaches the car safely. What is the maximum possible value of d? (3 Marks)
Ans. For the lion to catch the person over the distance d, the lion must reach the car at the same time as that of the person.
In the time t taken by the man to reach the car, the lion must travel a total distance of d + 26 m.
For the person: vman=d/t
Or, t = d/vman = d/4
For the lion: vlion = d/t
Or, t = d/vlion = d+26/6
d/4 = d+26/6
6d = 4d + 104
d = 52 m
Thus, the maximum possible distance is 52 m.
Ques. A projectile is projected at an angle > 45 degrees with an initial velocity of u. The time t, at which its horizontal velocity will be equal to vertical velocity is_________. (5 Marks)
Ans. The Angled Projectile Motion’s
Horizontal velocity component is given by:
vx = u cos (θ)
where v, is the horizontal velocity, u is the initial velocity and θ is the angle of the projectile.
Vertical velocity component is given by:
v = u sin (θ) - gt
where vy is vertical velocity,u is the initial velocity, θ is the angle of the projectile, g is the acceleration due to gravity and 't' is the time of the projectile.
The time 't' at which the horizontal and vertical velocity will be equal is given by
u cos = u sin (θ) - gt
gt = u sin (θ) - u cos (θ)
gt = u [sin (θ) - cos (θ)]
t = u [sin (θ) - cos (θ)] /g
Ques. Two trains each of the length 109 m and 91 m are moving in opposite directions with velocities 34 km h-1 and 38 km h-1 respectively. At what time the two trains will completely cross each other? (4 Marks)
Ans. Let l1, l2 be the lengths of the two trains.
v1, v2 be their velocities respectively.
∴ l1 = 109m, l2 = 91 m, v1 = 34kmh-1, v2 = 38kmh-1.
As the trains are moving in opposite directions so relative velocity of the trains is given by
v1 – (- v2) = v1 + v2
= 34 + 38 = 72 kmh-1
= 72 × 518 = 20 ms-1
Total distance to be covered by the two trains in crossing each other
= l1 + l2= 109 + 91 = 200 m
If t be the time taken in crossing, then t can be calculated using the relation
x = vt
or
t = 200/20 = 10s
Ques. Ambala is at a distance of 200 km from Delhi. Ram sets out from Ambala at a speed of 60 km h-1 and Sham sets out at the same time from Delhi at a speed of 40 km h-1. When will they meet? (4 Marks)
Ans.
S = 200 km. Let VR and vs be the speeds of Ram and Sham respectively moving in opposite directions.
∴ VR = 60 kmh-1, Vs = 40 kmh-1.
∴ Relative velocity of Ram w.r.t. Sham is
VRS = VR – (- VS)
= VR + VS
= 60 + 40 = 100 kmh-1
If t = time after which they will meet, then
t = time taken in covering 200 km distance with VRS
i.e. \(t = \frac{200}{V_{RS}}\) = \(\frac{200 km}{100 kmh^{-1}}\) = 2h.
∴ Time after which they meet = 2h.
Ques. A car traveling at a speed of 60 km h-1 on a straight road is ahead of a scooter traveling at a speed of 40 km h-1. How would the relative velocity be altered if the scooter is ahead of the car? (3 Marks)
Ans. vc = speed of car = 60 kmh-1
vs = speed of scooter = 40 kmh-1
vcs = relative velocity of car w.r.t. scooter
= vc – vs
= 60 – 40
= 20 kmh-1
Similarly Vsc = relative velocity of scooter w.r.t. car
= Vs – Vc
= 40 – 60
= – 20 kmh-1
Thus we conclude that the magnitude of the relative velocity is the same in both cases but the direction of relative velocity is reversed if the scooter is ahead of the car.
Ques. A ball is thrown vertically upward with velocity of 20 ms-1. It takes 4 seconds to return to its original position. Draw velocity-time graph for the motion of the ball and answer the following questions: (4 Marks)
At which point P, Q, R, the stone has:
(a) reached its maximum height.
(b) stopped moving?
Ans.
Let P represent the initial position at the time when the ball is thrown vertically upward.
Q represents the highest point reached by the ball.
R represents the original position of the ball after 4 seconds.
Thus, velocity-time graph for the motion of the ball is as shown below.
(a) We know that at the highest point, the velocity of the object is zero. So stone will reach its maximum height corresponding to point Q.
(b) The stone has stopped moving at point Q because at Q, v = 0.
Ques. What is always greater than or equal to the average velocity? (1 Mark)
Ans. Average speed is always greater than or equal to the average velocity.
Ques. What is the definition of constant velocity? (2 Marks)
Ans. A body occupies constant velocity if it has constant speed and a constant direction. Constant direction restricts the body to move in a straight trajectory path. Hence, constant velocity can also be defined with a body that moves in a straight path at a constant speed.
Ques. Define Velocity. (1 Mark)
Ans. The velocity of an object is the rate at which an object's position varies with a frame of reference and time. Velocity is often expressed as the displacement of an object in a unit time.
Ques. Show the purpose of the velocity-time graph. (2 Marks)
Ans. Velocity-time graph is usually used in order to explain the constant acceleration of an object.
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