The displacement $ x $ versus time graph is shown below.
The displacement $ x $ is plotted against time $ t $. Choose the correct answer from the options given below:
The displacement $ x $ versus time graph is shown below.
The displacement $ x $ is plotted against time $ t $. Choose the correct answer from the options given below:
Show Hint
- The average velocity during 0 to 3 s is 10 m/s
- The average velocity during 3 to 5 s is 0 m/s
- The instantaneous velocity at \( t = 2 \) s is 5 m/s
- The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s
The Correct Option is D
Approach Solution - 1
To solve this problem, we need to analyze the displacement versus time graph given and understand the definitions of average velocity and instantaneous velocity.
Definitions
- Average Velocity: It is defined as the total displacement divided by the total time taken. Mathematically, \(v_{\text{avg}} = \frac{\Delta x}{\Delta t}\).
- Instantaneous Velocity: It is the velocity of an object at a specific instant of time. It is obtained by finding the slope of the tangent to the graph at that point.
Analysis of Options
- The average velocity during 0 to 3 s is 10 m/s:
In this interval, the object moves from point A (at 0 s and \(x = -5 \, \text{m}\)) to point C (at 3 s and \(x = 5 \, \text{m}\)).
Total displacement = \(5 - (-5) = 10 \, \text{m}\).
Average velocity = \(\frac{10 \, \text{m}}{3 \, \text{s}} = \frac{10}{3} \, \text{m/s} \neq 10 \, \text{m/s}\).
This option is incorrect. - The average velocity during 3 to 5 s is 0 m/s:
From 3 s to 5 s, the object moves from point C (5 m) to point D (5 m), so there is no displacement.
Total displacement = \(5 - 5 = 0 \, \text{m}\).
Average velocity = \(\frac{0}{2} = 0 \, \text{m/s}\).
This option is correct, but let's check the other options. - The instantaneous velocity at \( t = 2 \) s is 5 m/s:
At \(t = 2 \, \text{s}\), the graph shows a linear segment from A to C with a slope of \(5 \, \text{m/s}\).
This option is consistent, but not the only correct one. - The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s:
From 5 s to 7 s, the object moves from point E (10 m at 5 s) to point G (0 m at 7 s).
Total displacement = \(0 - 10 = -10 \, \text{m}\).
Average velocity = \(\frac{-10}{2} = -5 \, \text{m/s}\).
At \( t = 6.5 \) s, the graph section EF has a slope of -5 m/s.
This option is correct and matches the provided correct answer.
Conclusion
The correct answer is: The average velocity during 5 to 7 s is the same as instantaneous velocity at \( t = 6.5 \) s.
Approach Solution -2
To find the average and instantaneous velocities, we use the formulas:
1. Average Velocity: \[ \bar{v} = \frac{\Delta x}{\Delta t} \] where \( \Delta x \) is the change in displacement and \( \Delta t \) is the time interval.
2. Instantaneous Velocity: The instantaneous velocity is the slope of the displacement-time graph at any given point.
Now, let's calculate each part:
- (A) Average velocity during 0 to 3 s: The displacement changes from \( 0 \) to \( 30 \, \text{m} \) in 3 seconds: \[ \bar{v} = \frac{30 - 0}{3 - 0} = \frac{30}{3} = 10 \, \text{m/s} \]
Hence, (A) is correct.
- (B) Average velocity during 3 to 5 s: The displacement does not change from \( t = 3 \) to \( t = 5 \) s, so: \[ \bar{v} = \frac{0 - 0}{5 - 3} = 0 \, \text{m/s} \] Hence, (B) is correct.
- (C) Instantaneous velocity at \( t = 2 \) s: From the graph, the slope of the line at \( t = 2 \) s is: \[ v = 5 \, \text{m/s} \] Hence, (C) is correct.
- (D) Average velocity during 5 to 7 s: The displacement changes from \( 10 \, \text{m} \) to \( 10 \, \text{m} \) (no change) in 2 seconds, so: \[ \bar{v} = \frac{10 - 10}{7 - 5} = 0 \, \text{m/s} \] The instantaneous velocity at \( t = 6.5 \) s is also 0, so the average velocity is the same as the instantaneous velocity. Hence, (D) is correct.
- (E) Average velocity from \( t = 0 \) to \( t = 9 \) s: The displacement returns to zero at \( t = 9 \) s, so the average velocity is: \[ \bar{v} = \frac{0 - 0}{9 - 0} = 0 \, \text{m/s} \] Hence, (E) is correct.
Thus, the correct answer is (4).
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Waves Questions
- A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm, both vibrating in fundamental mode. The velocity of sound is ________ m/s.
- The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60 cm, the length of the closed pipe will be :
- Two waves of intensity ratio \( 1 : 9 \) cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is \( I_1 \) (b) Waves are coherent is \( I_2 \) and differ in phase by \( 60^\circ \) If \( \frac{I_1}{I_2} = \frac{10}{x} \) then \( x = \) _____.
- The equation of a stationary wave is:
\[y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right)\]Which of the following is NOT correct: - A plane progressive wave is given by \( y = 2 \cos 2\pi(330t - x) \, \text{m} \). The frequency of the wave is:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.