The area under the velocity-time graph for a particle moving in a straight line with uniform acceleration gives
Understanding Area Under Velocity-Time Graphs
The question asks what physical quantity is represented by the area under the velocity-time graph for a particle moving in a straight line with uniform acceleration. Let's explore the relationship between velocity, time, and displacement.
What is a Velocity-Time Graph?
A velocity-time graph plots the velocity of an object on the vertical (y) axis against time on the horizontal (x) axis. The shape of the graph tells us about the object's motion. For a particle moving with uniform acceleration, the velocity-time graph is a straight line.
Area Under the Velocity-Time Graph
Consider a small time interval \(\Delta t\) on the velocity-time graph. During this small interval, if the velocity is approximately \(v\), then the displacement during this time is approximately \(v \times \Delta t\). This is essentially the area of a narrow rectangle under the graph for that time interval.
To find the total displacement over a larger time interval, we can sum up the areas of all such small rectangles. In the limit as \(\Delta t\) approaches zero, this sum becomes an integral. The area under the velocity-time graph is given by the definite integral of velocity (\(v\)) with respect to time (\(t\)):
\(\text{Area} = \int_{t_1}^{t_2} v(t) \, dt\)
By definition, the integral of velocity with respect to time gives the displacement of the particle. Displacement is the change in position of the particle.
Analyzing the Options
Let's look at the given options:
- its average velocity: Average velocity is defined as the total displacement divided by the total time taken. The area under the curve represents total displacement, not average velocity directly.
- its net displacement: As explained above, the area under the velocity-time graph precisely represents the net change in position, which is the net displacement, during the time interval considered.
- the distance travelled by it: Distance travelled is the total length of the path covered, regardless of direction. While the magnitude of displacement equals the distance travelled when the velocity does not change direction (i.e., the velocity is always positive or always negative), if the velocity-time graph crosses the time axis (meaning the velocity changes direction), the area above the axis represents displacement in one direction and the area below represents displacement in the opposite direction. The net displacement is the sum of these signed areas. The total distance travelled would be the sum of the magnitudes of these areas (the area under the absolute value of the velocity). The area under the velocity-time graph gives displacement, not necessarily distance travelled, especially if there is a change in direction.
- its average speed: Average speed is the total distance travelled divided by the total time taken. Like average velocity, the area under the graph does not directly give average speed.
Therefore, the area under the velocity-time graph for a particle moving in a straight line gives its net displacement.
Revision Table: Velocity-Time Graph Concepts
| Feature of Velocity-Time Graph | Physical Quantity Represented | How to Calculate/Interpret |
|---|---|---|
| Slope | Acceleration | Change in velocity divided by change in time (\(\frac{\Delta v}{\Delta t}\)) |
| Area under the graph | Displacement (net) | Integral of velocity over time (\(\int v \, dt\)) |
| Position on graph | Velocity at that instant | Value on the y-axis at a given time |
| Intercept on y-axis | Initial velocity | Velocity at time t=0 |
Additional Information on Motion Graphs
Understanding different types of motion graphs is crucial for kinematics. Besides velocity-time graphs, we also study position-time graphs and acceleration-time graphs.
- Position-Time Graph: The slope of a position-time graph gives the instantaneous velocity. The area under a position-time graph does not represent a standard physical quantity.
- Acceleration-Time Graph: The area under an acceleration-time graph gives the change in velocity. The slope of an acceleration-time graph represents jerk (rate of change of acceleration).
For motion in a straight line, displacement, velocity, and acceleration are vectors, meaning they have both magnitude and direction. Distance and speed are the magnitudes of displacement and velocity, respectively, and are scalar quantities.
Similar Questions
A particle moves with uniform acceleration along a straight line from rest. The percentage increase in displacement during the sixth’ second compared to that in the fifth second is about
An electron and a proton starting from rest are accelerated through a potential difference of 1000 V. Which one of the following statements in this regard is correct?
An athlete completes one round of a circular track of diameter 100 m in 20 s. What will be the displacements after 1 minute and 10 s, respectively ?
A boy completes one round of a circular track of diameter 200 m in 30 s. What will be the displacement at the end of 3 minutes and 45 seconds?
The direction of acceleration in uniform circular motion is along the
A car moving with a speed of 12 m/s is subjected to brakes which produces a deceleration of 6 m/s2 The car takes 2s to stop after the application of brakes. What is the distance covered by the car after the application of brakes?
Important Questions from Motion
An object is covering distance in direct proportion to the square of time elapsed. What conclusion can be drawn about the motion of the object?
If the distance time graph of the motion of an object is a straight line but not parallel to the time axis, then it may be concluded that the object is moving with a:
Which of the following changes when a body performs uniform circular motion?
Vehicles have treaded tires so that it_______.
If the initial velocity of an object thrown upwards is 14 m/s, then the time taken for the object to reach its highest point will be_______. (a = 9.8 m/s2)
