**Graphical Representation of Motion:Derivation of Equation of Motion**

While studying motion of bodies we have to keep two things in mind. We have to study various characteristics associated with moving bodies while we are to explore and study in detail, about the cause of producing motion. **Kinematics **is the branch of physics which deals only with the description of motion of bodies. Mainly there are two type of graphs are belonging to the kinematics. One is position-time graph and the other is velocity-time graph.

These graphs will have validity only if motion under study is along a straight line. Then, displacement, velocity and acceleration vectors are collinear and can be treated as algebraic quantities. Let x-axis be the path of motion. Then, x-coordinate represents the magnitude of position vector.

**Position - Time Graph : ****Derivation of Equation of Motion**

If we plot time t along the x-axis and the corresponding position (say x) from the origin O on the y-axis, we get a graph which is called the position-time graph. This graph is very convenient to analyse different aspects of motion of a particle. Let us consider the following cases.

(i) In this case, position (x) remains constant but time changes. This indicates that the particle is stationary in the given reference frame. Hence, the straight line nature of position-time graph parallel to the time axis represents the state of rest. Note that its slope (tan Î¸) is zero.

(ii) When the x-t graph is a straight line inclined at some angle (Î¸≠) with the time axis, the particle traverses equal displacement Î”x in equal intervals of time Î”t. The motion of the particle is said to be uniform rectilinear motion. The slope of the line measured by Î”x/Î”t = tÎ¸ represents the uniform velocity of the particle.

(iii) When the x-t graph is a curve, motion is not uniform. It either speeds up or slows down depending upon whether the slope (tan Î¸ successively increases or decreases with time. As shown in the figure the motion speeds up from t = 0 to t=t1 (since the slope tan Î¸ increases). From t=t1 to t=t2, AB represents a straight line indicating uniform motion. From t=t2 to t=t3, the motion slows down and for t>t3 the particle remains at rest in the reference frame.

**The acceleration-time graph:****Derivation of Equation of Motion**

Acceleration time curves give information about the variation of acceleration with time. Area under the acceleration time curve gives the change in velocity of the particle in the given time interval.

**Did You Know **

- The displacement can have positive, negative or zero value.
- The displacement is never greater than the actual distance travelled.
- A change in either speed or direction of motion results in a change in velocity.
- It is not possible for a particle to possess zero speed with a non-zero velocity.
- The velocity-time graph for a uniformly accelerated body is a straight line.

**Question 1:-**

The position-time graph of a body is parallel to t-axis. The body is said to be moving with:

(a) uniform velocity (b) non-uniform velocity

(c) may be uniform or npn-uniform velocity **(d) no motion at all**

**Question 2:-**

The relation between time t and distance x is t = ax2+bx, where a and b are constants. The acceleration is:

(a) 2bV3 (b) -2abV2

(c) 2aV2 ** (d) -2aV3**

**Question 3:-**

A car has to cover the distance 60 km, if half of the total time it travels with velocity 80 km/h and in rest half time its speed becomes 40 km/h, the average speed of car will be,

**(a) 60 km/h ** (b) 80 km/h

(c) 120 km/h (d) 180 km/h

**Question 4:-**

In a straight line motion the distance travelled is propertional to the square root of the time taken. The acceleration of the particle is proportional to:

(a) velocity (b) V2

**(c) V3 ** (d) √V

**Question 5:-**

The displacement of a particle along a straight line at time t is given by x = b0+b1t+b2t2. The acceleration of the particle is:

(a) b0 (b) b1

**(c) 2b2 ** (d) b2/2