Velocity-Time Graphs for One-Dimensional Motion
For one-dimensional motion, the velocity ($v$) of a particle must be a unique function of time ($t$). This means that for any specific instant in time, there can only be one corresponding velocity value. Graphically, this implies that any vertical line drawn on a velocity-time ($v-t$) graph must intersect the curve at most once.
Analysis of Graphs
- Graph 1 (Image A): This graph depicts velocity as a function of time. For every time value shown, there is only one velocity value. This is physically possible for one-dimensional motion.
- Graph 2 (Image B): This graph shows multiple velocity values for a single time value (at the peak of the curve). This violates the definition of a function and cannot represent the motion of a single particle.
- Graph 3 (Image C): Similar to Graph 2, this graph is not a valid function, as it shows multiple velocity values for certain time instances, particularly where the velocity appears to change direction abruptly.
- Graph 4 (Empty): This option is represented by an empty tag. If this represents a particle at rest ($v=0$ for all $t$), it is a valid case of one-dimensional motion.
Conclusion
A valid $v-t$ graph must represent velocity as a unique function of time. Graph 1 satisfies this condition. Assuming the empty Graph 4 represents a particle at rest ($v=0$), it also represents valid one-dimensional motion. Graphs 2 and 3 are invalid because they do not represent velocity as a unique function of time.
Therefore, Graph 1 and Graph 4 can possibly represent one-dimensional motion.
Similar Questions
- A body of density '$\rho$' is dropped slowly on the surface of a lake of depth $d$. If the density of the lake water be '$\rho'$' ($\rho' < \rho$) then the time taken by the body to reach the bottom of the lake is
Three blocks of masses $m_1 = 2\text{ kg}$, $m_2 = 3\text{ kg}$ and $m_3 = 5\text{ kg}$ are placed on a horizontal frictionless surface and a force of 30N pulls the system as shown below. The value of tension in the string between $m_2$ and $m_3$ will be
- From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken by the particle to hit the ground is $n$ times that taken by it to reach the highest point of its path. The relation between $H$, $u$ and $n$ is
A body initially at rest and sliding along a frictionless track from a height '$h$' (as shown in figure) just completes a vertical circle of diameter AB = $d$. The height '$h$' is equal to
A uniform rod $AB$ is suspended from a point $P$, at a variable distance $x$, from $A$, as shown in figure. To make the rod horizontal, a mass '$m$' is suspended from its end $A$. Which set of variables will give a straight line when they are plotted?
A particle of mass $m$ is suspended from a point O by a string of length $R$. It is given a velocity $u = 3\sqrt{gR}$ at the bottom. The difference in tension at point $B$ and at the point $C$ is
The moment of inertia of a thin disc about axes $a, b, c, d$ are $I_{1}, I_{2}, I_{3}$ and $I_{4}$ respectively, as shown in figure. If the moment of inertia about an axis passing through the centre and perpendicular to the plane of the disc is $I$ then,
A train is moving along the tracks at a constant speed u. A girl on the train throws a ball of mass m straight ahead along the direction of motion of the train with speed v with respect to herself. Then
An earth’s satellite near the surface of the earth takes about 90 min per revolution. A satellite orbiting the moon also takes about 90 min per revolution. Then which of the following is true ?
[where ρm is density of the moon and ρe is density of the earth.]
There are n elastic balls placed on a smooth horizontal plane. The masses of the balls are m, \(\dfrac{m}{2},\dfrac{m}{2^{2}},....\dfrac{m}{2^{n-1}}\) respectively. If the first ball hits the second ball with velocity v0, then the velocity of the nth ball will be,
Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
