Instantaneous Velocity Formula: Definition and Solved Examples

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Instantaneous Velocity is the speed of a moving object at a given time, maintaining a specific direction. The velocity of an object changes with change in time. Instantaneous velocity is the measurement of velocity as the difference in time reaches zero on the other hand velocity is the ratio of change in position and change in time. In the following sections, we will discuss the instantaneous velocity formula with solved examples.

Read Also: Motion in a Straight Line

Table of Content

Instantaneous Velocity
Concept of Instantaneous Velocity
Formula of Instantaneous Velocity
Derivation of Instantaneous Velocity Formula
Things to Remember
Important Questions

Instantaneous Velocity

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Instantaneous velocity is the velocity of an object in motion at a specific point in time.It is a vector quantity and has both magnitude and direction. Instantaneous velocity is expressed in m/s.

X is the given function concerning the time t.

Where,

Δt is the small-time interval.
V(t) is the instantaneous velocity at time t
x is the displacement.
t is the time.

Motion in a Straight Line Video Explanation

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Concept of Instantaneous Velocity Formula

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The rate of change of a position for a negligible time interval is known as instantaneous velocity. Instantaneous velocity is similar to instantaneous speed with only one difference that instantaneous speed has no direction. Thus, instantaneous velocity can be defined as the velocity of a moving object at a specific point in time.

The needle of the speedometer which shows the speed of the car every hour fluctuates. This fluctuation along with the direction of the car in a particular period is known as instantaneous velocity.

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Formula for Instantaneous Velocity

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It is determined as the average velocity with a minimum period. The average velocity can be calculated as the ratio of total displacement and total time.

The time interval is directly proportional to the displacement. Thus, when displacement reduces to zero, the time interval also becomes the same. But there is a value to the limit of the ratio between time and displacement which is known as instantaneous velocity.

It can also be represented by taking the slope of the distance-time graph or x-t graph. The formula of instantaneous velocity is:

V(t) = dx(t)/ΔT

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Derivation of the formula

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Instantaneous velocity is the velocity between the two points in a time limit where the time between the two points gradually becomes zero. X(t) is the position of x in respect to the function of t. The equation for average velocity between the two points is:

V= [x(t₂) – x(t₁)]/(t₂-t₁)

Let, t₁ = t and t₂ = t + \(\Delta\)t

To calculate instantaneous velocity, the \(\Delta\)t needs to be zero.

Taking the limit as \(\Delta\)t -> 0 and inserting the expressions in the equation,

Instantaneous velocity or v(t) = Lim [x(t+\(\Delta\)t) – x(t)]/\(\Delta\)t

Thus, v (t) = dx(t)/dt

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Things to Remember

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Instantaneous velocity is the velocity of an object in motion at a specific point in time. The SI unit of instantaneous velocity is m/s.
It is a vector quantity and has both magnitude and direction.
The formula of instantaneous velocity is- V= [x(t₂) – x(t₁)]/(t₂-t₁)
Instantaneous velocity can be calculated and represented graphically as well as numerically
At all instants or time intervals, average velocity and velocity is the same in the case of uniform motion
Instantaneous speed affects the intensity of instantaneous velocity.
Instantaneous velocity is a vector quantity
Average velocity = \(\Delta\)S/\(\Delta\)T where, \(\Delta\)S = distance covered and \(\Delta\)T = the time interval

Important Questions

Ques: A specific object is moving in a straight line for time (t) = 3s with a function x = 5t² + 2t + 3. Calculate the instantaneous velocity. (2 Marks)

Ans: vinst = dx/ dt

= d(5t²+2t+3)/dt

Vinst = 10t +2

t = 3s ,

Instantaneous velocity v (t) = 10t +2

v(3) = 10(3) +2

v(3) = 32m/s

Ques: Calculate the instantaneous velocity at t= 2.0s when the position of the given particle is x(t)=3.0t + 0.5t3m. (2 Marks)

Ans: x(t)=3.0t + 0.5t3m

V(t) = dx(t)/dt = 3.0 + 1.5t² m/s

Given t= 2.0s,

V(2.0s ) = [3.0 + 1.5(2.0)²] m/s

= 9.0 m/s

Ques: The equation of motion of a bullet travelling in a straight line is S(t)=4t + 6t² which travels for 15 seconds before crushing. Calculate the instantaneous velocity at the time interval of 10th second. (3 Marks)

Ans: S(t)=4t + 6t²

ds/dt = d(4t + 6t²)/dt

=4 +12t

Therefore, vinst at (t=10) = 4 + (12x10)

= 124 m/s

Ques: The approximate equation of motion of a body under gravity is S(t)= 4.9t². calculate the instantaneous velocity at 5 seconds after the release. (2 Marks)

Ans: S(t)= 4.9t²

Instantaneous velocity at 5 seconds,

Vinst = [ds/dt]t=5 = 4.9 x 2 x5

= 49m/s

Ques: The maximum height of a rocket in the first 30 mins of its launch is given by x=bt² when b is constant with the value of 2.9m/s². Calculate the general expression for the velocity of the rocket as a function of time and from it the instantaneous velocity at the given time (t) = 20 seconds. (3 Marks)

Ans: velocity (v) = dx/dt

Here, x = bt²

V = dx/dt = d(bt²)/dt = 2bt

General expression for the velocity of rocket (V) = 2bt …… (i)

instantaneous velocity at t = 20 secs ,

2bt = 2 x 2.9 x 20 = 116m/s …..(ii)

Instantaneous velocity = 116m/s

Ques: X= a + bt² is the position of an object moving along the x-axis where a= 8.5m, b = 2.5ms^-2. Calculate the velocity at t= 0s and t= 2.0s. also calculate the average velocity between t= 2.0s and t = 4.0s. (3 Marks)

Ans: velocity (v) = dx/dt = d(a +bt²)/ dt

= 2bt

= 5.0 tms^-1

At t = 0s, v = 0ms^-1 and at t = 2.0s, v = 10ms^-1

Average velocity = x (4.0) – x(2.0)4.0-2.0

= a+16b-a-4b².0 = 6.0 x b

= 6.0 x 2.5

= 15ms^-1

Ques: The position of a moving car is given by the function x = 4t2 + 10t + 6. Calculate the instantaneous velocity at the time (t) = 5 s (3 Marks)

Ans: x = 4t2 + 10t +6

Vinst = dx/dt

= d (4t2+ 10t + 6)/dt

Given, t= 5s

V(t) = 8t+10

V(5) = 8(5) +10

V(5) = 50m/s

Ques: The position of an object is given by x(t) = 3.0t + 0.5t³m. calculate the instantaneous velocity at t=2.0s and the average velocity between 1.0s and 3.0s (3 Marks)

Ans: v(t) = dx(t)/dt = 3.0 + 1.5t² m/s

Given t=2.0s

V(2.0s) = [3.0 + 1.5(2.0)2 ] m/s

= 9.0m/s

Average velocity between 1.0s and 3.0s , we calculate the values of x(1.0s) and x(3.0s):

X(1.0s) = [(3.0)(1.0) + 0.5 (1.0)3 ] m = 3.5m

X(3.0s) = [(3.0)(3.0) + 0.5(3.0)3]m = 22.5m

The final average velocity is,

V = x3.0s-x(1.0s)t³.0s-t(1.0s) = 22.5-3.5m³.0-1.0s

= 22.5-3.5m³.0-1.0s

= 9.5 m/s

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CBSE CLASS XII Related Questions

1.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:
- \( \frac{q}{4 \pi \epsilon_0 l^2} \) pointing along MA
- \( \frac{q}{\pi \epsilon_0 l^2} \) pointing along AM
- \( \frac{q}{2 \pi \epsilon_0 l^2} \) pointing along AM
- Zero
View Solution
2.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))
View Solution
3.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]
View Solution
4.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.
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5.
If Bohr’s quantization postulate (angular momentum \( = \frac{nh}{2\pi} \)) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why, then, do we never speak of quantization of orbits of planets around the Sun? Explain.
View Solution
6.
Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).
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Instantaneous Velocity Formula: Definition and Solved Examples

Instantaneous Velocity

Motion in a Straight Line Video Explanation

Concept of Instantaneous Velocity Formula

Formula for Instantaneous Velocity

Derivation of the formula

Things to Remember

Important Questions

CBSE CLASS XII Related Questions

1.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

2.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5.
If Bohr’s quantization postulate (angular momentum \( = \frac{nh}{2\pi} \)) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why, then, do we never speak of quantization of orbits of planets around the Sun? Explain.

6.
Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).

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Instantaneous Velocity Formula: Definition and Solved Examples

Instantaneous Velocity

Motion in a Straight Line Video Explanation

Concept of Instantaneous Velocity Formula

Formula for Instantaneous Velocity

Derivation of the formula

Things to Remember

Important Questions

CBSE CLASS XII Related Questions

1.The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

2.Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4. If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5. If Bohr’s quantization postulate (angular momentum \( = \frac{nh}{2\pi} \)) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why, then, do we never speak of quantization of orbits of planets around the Sun? Explain.

6.Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).

Comments

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1.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

2.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5.
If Bohr’s quantization postulate (angular momentum \( = \frac{nh}{2\pi} \)) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why, then, do we never speak of quantization of orbits of planets around the Sun? Explain.

6.
Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).