Question

What is the velocity-time graph if a body is in motion with variable acceleration?
${\text{A}}{\text{.}}$ is a straight line with a positive slope
${\text{B}}{\text{.}}$ is a straight line with a negative slope
${\text{C}}{\text{.}}$ is a straight line parallel to the time axis
${\text{D}}{\text{.}}$ is a curve

Answer 1

- Hint- Here, we will proceed by defining the terms velocity and acceleration used in kinematics. Then, we will use the concept that slope of the velocity-time graph is simply the acceleration.

Complete step-by-step solution -

Velocity is a vector quantity that indicates displacement, time, and direction. Unlike speed, velocity measures displacement, a vector quantity indicating the difference between an object's final and initial positions. Speed measures distance, a scalar quantity that measures the total length of an object's path.
Acceleration is defined as a vector quantity that indicates the rate of change of velocity. It has dimensions of length and time over time. Acceleration is often referred to as "speeding up", but it really measures changes in velocity.
The slope of a velocity graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
The slope of a velocity-time graph will be given by the following formula:
Slope = $\dfrac{{{\text{Final velocity}} - {\text{Initial Velocity}}}}{{{\text{Time taken}}}} = \dfrac{{{\text{Change in velocity}}}}{{{\text{Change in time}}}}$
Acceleration is simply the slope of the velocity-time graph.
If the body is in uniform motion (no change in velocity or acceleration is zero), then the velocity-time graph is a horizontal straight line (i.e., the slope is zero). If the body is in motion with constant positive acceleration (i.e., velocity is increasing with time), then the velocity-time graph is a straight line with upward sloping (i.e., the slope is positive). If the body is in motion with constant negative acceleration (i.e., velocity is decreasing with time), then the velocity-time graph is a straight line with downward sloping (i.e., the slope is negative). If the body is in motion with variable acceleration (i.e., acceleration is changing with time), then the velocity-time graph is a curve of any shape because the slope of the velocity-time graph (i.e., acceleration) is different at different points.
Therefore, option D is correct.

Note- Speed is a scalar quantity that indicates the rate of motion distance per time whereas velocity is a vector quantity. A quantity is said to be scalar if only magnitude is defined for that quantity whereas a quantity is said to be vector if both magnitude and direction are defined for that quantity.