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# Graphs: Position-Time, Velocity-Time, Inter-Conversion, Practice Problems and FAQs

You ride to school by bike everyday. You start from rest, pedal faster, reach a certain velocity say 5 ms-1, in 10 s. Would you be able to find the displacement in 10 s? Short answer : yes, if you applied equations of motion. Another approach would be to take a pen and paper,  plot a simple graph with velocity on one axis and time on the other; the area under such a  graph would give displacement. On the other hand, plotting graphs of displacement x and time t  gives us an idea about the velocity.

A graph plotted between velocity V on the y - axis and time t on the  x - axis  is called velocity time graph or  v - t graph. Similarly, a graph plotted between position x on the vertical axis and time x on the horizontal axis is called position- time graph. With the information of v - t graph, both displacement and acceleration can be calculated. Before we dive into the details, one must understand two vital terms: slope and area. Slope is defined as the tangent of the angle which the line or tangent to the curve makes with the +Ve direction of x - axis; in the following graph for instance; slope of the curve

slope of x-t curve gives velocity

Area under the graph is the area the curve encloses with x - axis. Area under the v - t graph gives the displacement of the body

Area under the v-t curve gives displacement

## Displacement Time Graph

•  Graph plotted between position/displacement on the y - axis and time t on the x - axis is called position/displacement - time or x - t graph.
• Slope of the x - t   graph gives velocity.

The x - t graph for various cases is shown.

Note: If acceleration is a function of position or time, then such acceleration is said to be non uniform.

## Velocity Time Graph

• A graph plotted between velocity v on the y - axis and time t on the  x - axis is called the v - t graph.
• Slope of the v - t graph gives acceleration
• Area under the v - t graph gives displacement.

The v - t graph for different cases is as shown.

Note:

• The mathematical interpretation of slope of a graph is its derivative .i.e differentiating displacement x wrt time t gives velocity. The same applies for acceleration; , differentiating velocity v wrt time t gives acceleration a.
•  Integration refers to finding area under the graph. For example;

i.e area under v - t graph gives the displacement.

## Practice Problems of Graphs

Question 1. A person walks 10 m in the positive direction in 5 seconds, then stops for 2  seconds, and then walks 15 m in the negative direction for 5 seconds. Draw his displacement-time graph.

Answer:  The motion can be summarized as follows:

(i) 10 m in the positive direction from 0 to 5 seconds.

(ii) At rest from t = 5 to t = 7 s

(iii)15 m in the negative direction from t = 7 to t = 12 s

The following diagram represents his motion

Question 2. The x - t (position time graph) of an object moving in a straight line is shown. Calculate the average velocity in the time interval

(i)t = 2 to t = 4 s

(ii)t = 4s to t = 6 s

Question 3. Given below is a  v - tgraph, find the (i) displacement in first 3  seconds (ii) acceleration in first 3 seconds.

Question 4. From the displacement time graph given below, find velocity?

## FAQs of Graphs

Question 1. Draw the v - t graph for a stone thrown upwards;

Answer: Acceleration a = - g for a stone thrown up;

Question 2. Draw the v - t graph for a ball thrown downwards from the top of a tower.

Answer: For a ball thrown downwards;

Question 3. How to find acceleration from v - t graph?
Slope of the v - t graph gives acceleration a.

Question 4.  How to find displacement from v - t graph?
Area under the v - t graph gives displacement.

## NCERT Class 11 Physics Chapters

 Physical World Units and Measurements Motion in a Straight Line Motion in a Plane Laws of Motion Work Energy and Power Particles and Rotational Motion Gravitation Mechanical Properties of Solids Mechanical Properties in Liquids Thermal Properties of Matter Thermodynamics Kinetic Theory Oscillations Waves

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