Nature of acceleration in one dimensional motion One dimensional motion results from the action of net external force that applies along the direction of motion. It is a requirement for motion to be in one dimension. In case, force and velocity are at certain angle to each other, then there is sideway deflection of the object and the resulting motion is no more in one dimension. If velocity and force are in the same direction, then magnitude of velocity increases; If velocity and force are in the opposite direction, then magnitude of velocity decreases. The valid combination (i and ii)and invalid combination (iii) of velocity and acceleration for one dimensional motion are shown in the figure.

Acceleration – time plot The requirement of one dimensional motion characterizes the nature of acceleration involved. The acceleration may vary in magnitude only. No sideway directional change in acceleration of the motion is possible for a given external force. We must emphasize that there may be reversal of motion i.e. velocity even without any directional change in acceleration. A projectile, thrown up in vertical direction, for example, returns to ground with motion reversed at the maximum height, but acceleration at all time during the motion is directed downwards and there is no change in the direction of acceleration.

Motion under gravity In mathematical parlance, if v = Ai, then a = Bi, where A and B are positive or negative numbers. For one dimensional motion, no other combination of unit vectors is possible. For example, acceleration can not be a = Bj or a = B(i + j). We summarize the discussion as : The velocity and force (hence acceleration) are directed along a straight line. For a given external force, the direction of acceleration remains unchanged in one dimension.

Graphs of one dimensional motion Graphs are signature of motion. Here, we shall discuss broad categories of motion types in terms of acceleration and velocity.

Acceleration – time plot Acceleration can be zero, constant or varying, depending upon the net external force and mass of the body. It is imperative that a single motion such as the motion of a car on the road may involve all kinds of variations in acceleration. A representation of an arbitrary real time acceleration - time variation may look like :

Acceleration – time plot We can interpret this plot if we know the direction of velocity. We consider that positive direction of velocity is same as that of acceleration. Section, A, in the figure, represents constant acceleration. Section B represents an increasing magnitude of acceleration, whereas section C represents a decreasing magnitude of acceleration. Nonetheless, all of these accelerations result in the increase of speed with time as both velocity and acceleration are positive (hence in the same direction). The section E, however, represents deceleration as velocity (positive) and acceleration (negative) are in opposite direction. There is no acceleration during motion corresponding to section D. This section represents uniform motion. For section A : Constant acceleration : a = đ v đ t = Constant For section B and C : Positive acceleration : a = đ v đ t > 0 For section D : Zero acceleration : a = đ v đ t = 0 For section E : Negative acceleration : a = đ v đ t < 0

Area under acceleration - time plot We know that : a = đ v đ t ⇒ đ v = a đ t Integrating both sides, we have : Δ v = a Δ t Thus, areas under acceleration – time plot gives the change in velocity in a given interval.

Velocity – time plot Here, we discuss velocity – time plot for various scenarios of motion in one dimension : 1: Acceleration is zero If acceleration is zero, then velocity remains constant. As such the velocity time plot is a straight line parallel to time axis. a = đ v đ t = 0 ⇒ đ v = 0 ⇒ v = Constant

Velocity – time plot 2: Acceleration is constant Constant acceleration means that instantaneous acceleration at all points during motion is same. It, therefore, implies that instantaneous acceleration at any time instant and average acceleration in any time interval during motion are equal. a avg = a Velocity of the object under motion changes by an equal value in equal time interval. It implies that the velocity – time plot for constant acceleration should be a straight line. Here, đ v đ t = constant = a ⇒ đ v = a đ t Integrating both sides, ⇒ v = a t + u This is a linear equation in time “t” representing a straight line, where “a” is acceleration and is equal to the slope of the straight line and "u" is the intercept on velocity axis, representing velocity at t = 0. The plot of velocity with respect to time, therefore, is a straight line as shown in the figure here.

Velocity – time plot For constant deceleration, the velocity – time plot has negative slope. Here, speed decreases with the passage of time :

Velocity – time plot Problem : A velocity – time plot describing motion of a particle in one dimension is shown in the figure.

Velocity – time plot Determine (i) Whether the direction of velocity is reversed during motion? (ii) Does the particle stop? (iii) Does the motion involve deceleration and (iv) Is the acceleration constant? Solution : (i) We see that velocity in the beginning is negative and changes to positive after some time. Hence, there is reversal of the direction of motion. (ii) Yes. Velocity of the particle becomes zero at a particular time, when plot crosses time axis. This observation indicates an interesting aspect of reversal of direction of motion (i.e. velocity): A reversal of direction of a motion requires that the particle is stopped before the direction is reversed. (iii) Yes. Occurance of decelration is determined by comparing directions of velocity and acceleration. In the period before the particle comes to rest, velocity is negative, wheras acceleration is always positive with respect to the positive direction of velocity (slope of the line is positive on velocity -time plot). Thus, velocity and acceleration are in opposite direction for this part of motion and is decelerated. Further, it is also seen that speed of the particle is decreasing in this period.(iv) The plot is straight line with a constant slope. Thus, motion is under constant acceleration. 3: The magnitude of acceleration is increasing Since slope of velocity – time curve is equal to the acceleration at that instant, it is expected that velocity – time plot should be a curve, whose slope increases with time as shown in the figure.

Velocity – time plot Here, a 1 < a 2 < a 3 4: The magnitude of acceleration is decreasing The velocity – time plot should be a curve, whose slope decreases with time as shown in the figure here :

Velocity – time plot Here, a 1 > a 2 > a 3 Problem : A person walks with a velocity given by |t – 2| along a straight line. Find the distance and displacement for the motion in the first 4 seconds. What are the average velocity and acceleration in this period? Discuss the nature of acceleration. Solution : Here, velocity is equal to the modulus of a function in time. It means that velocity is always positive. The representative values at the end of every second in the interval of 4 seconds are tabulated as below :


----------------------------
Time (s)      Velocity (m/s)
----------------------------
   0	          2
   1	          1
   2	          0
   3	          1
   4	          2
----------------------------

An inspection of the values in the table reveals that velocity linearly decreases for the first 2 second from 2 m/s to zero. It, then, increases from zero to 2 m/s. In order to obtain distance and displacement, we draw the plot between displacement and time as shown.

Velocity – time plot Now, area under the plot gives displacement. Displacement = Δ OAB + Δ BCD ⇒ Displacement = 1 2 x OB x OA + 1 2 x BD x CD ⇒ Displacement = 1 2 x 2 x 2 + 1 2 x 2 x 2 = 4 m We see that velocity does not change its sign. Thus, motion is unidirectional apart from being rectilinear. As such, distance is equal to displacement and is also 4 m. Now, average velocity is given by : v avg = Displacement Time = 4 4 = 1 m / s Similarly, average acceleration during the motion is : v avg = Δ v Δ t = 0 4 = 0 Though average acceleration is zero, the instantaneous acceleration of the motion during whole period is not constant as the velocity – time plot is not a single straight line. An inspection of the plot reveals that velocity – time has two line segments AB and BC, each of which separately represents constant acceleration. Magnitude of constant acceleration (instantaneous) in the first part of the motion is equal to the slope of the displacement – time plot AB : a AB = AO OB = -2 2 = - 1 m / s 2 Similarly, acceleration in the second part of the motion is : a AB = DC BD = 2 2 = 1 m / s 2 Thus, the acceleration of the motion is negative for the first part of motion and positive for the second of motion with respect to velocity on velocity-time plot. We can, therefore, conclude that acceleration is not constant.