Velocity-Time Curve Area and Displacement
The relationship between velocity (\(v\)), displacement (\(s\)), and time (\(t\)) is fundamental in kinematics. Velocity is defined as the rate of change of displacement with respect to time:
\( v = \frac{ds}{dt} \)
To find the change in displacement (\( \Delta s \)) over a time interval from \(t_1\) to \(t_2\), we can rearrange the formula:
\( ds = v \, dt \)
Integrating both sides over the interval \([t_1, t_2]\) gives the displacement:
\( \Delta s = \int_{t_1}^{t_2} v(t) \, dt \)
The definite integral \( \int_{t_1}^{t_2} v(t) \, dt \) geometrically represents the area under the velocity-time (\(v\)-\(t\)) curve between the times \(t_1\) and \(t_2\). Therefore, the area under the velocity-time curve corresponds to the displacement of the object during that time interval. Since the option provided is "magnitude of displacement", this confirms the concept.
Key takeaway: Area under v-t curve = Displacement.
Similar Questions
- An object starts from rest at x = 0 m and t = 0 s. It moves with a constant acceleration of \(2 \text{ m/s}^2\) along the x-axis. What is its average velocity between time 1 s and 5 s?
- A train travels 45 m in 3 s. What is the speed of the train in km/h?
- A ball is dropped from a height and falls freely for 2 seconds. Its average speed during this time is: (Consider: acceleration due to gravity =\(10 \text{ m/s}^2\))
Important Questions from Speed and Velocity
Which of the following quantities specifies its speed with direction?
The basic unit of speed of an object is ______.
- Which of the following represents non-uniform linear velocity?
- Which of the following describes the nature of the average speed?
- An object starts from rest at x = 0 m and t = 0 s. It moves with a constant acceleration of \(2 \text{ m/s}^2\) along the x-axis. What is its average velocity between time 1 s and 5 s?
