Theory

Distance & Displacement
Distance is the length travelled by a moving object and is a scalar quantity. Displacement is change in the position of a moving object, in other words it is the distance moved in a particular direction. It is therefore a vector quantity having both magnitude and direction.

Speed & Velocity
Speed of an object is the distance travelled by an object divided by the time taken to travel. Speed is a scalar quantity having only magnitude. Speed only tells us how fast or slow the object is moving, but not the direction it is travelling in. Velocity on the other hand, can be defined as the rate of change of displacement. You can think of velocity as ‘speed plus direction’. Consider two points say A and B. The time taken by an object to move from point A to point B can be defined as velocity and can be represented as,

v = dx/dt i.e. (small distance covered in a particular direction) / (small change in time)

In the above equation of velocity, as the small time taken approaches zero, we have the ‘instantaneous velocity’ of that object. The average velocity of an object can be defined as the total distance travelled by a body in a given direction divided by the total time taken.

Velocity is a vector quantity, so the velocity of an object can change if either its speed or direction changes. In other words, the velocity of a car travelling at constant speed changes when it makes a right turn. If the body is travelling in the same direction at a constant speed, it is then moving with a uniform velocity.

Acceleration
Acceleration can be defined as the rate of change of velocity. Acceleration determines how much the velocity of a body changes in every subsequent time interval. It is a vector quantity having a magnitude and direction. Any motion in which speed or direction varies is called accelerated motion.

Consider a body is moving with a constant velocity of 5 m/s. The time taken for the same body to move with a velocity of 10 m/s is known as acceleration. It can be represented as,

a = dv/dt i.e. (small change in velocity) / (small change in time)

The unit of acceleration is metres per second squared (m/s2) while velocity is measured in terms of metres per second (m/s).

linear_motion
   
Motion Graphs
 

The above figure shows two cars moving in the same direction.

The blue car has no acceleration and is therefore moving with a uniform velocity of 30 m/s. The second car has the same velocity of 30 m/s at point A, but is moving with a uniform acceleration of 10 m/s2.

The trails left by the blue car are spaced evenly indicating that this car is moving with a uniform velocity. The green car, however is accelerating and therefore the distance between the trails left by this car is keeps increasing as the car moves. The same observations can be made from the velocity plots for both the car. The Acceleration graph denotes zero acceleration for the blue car while the straight horizontal green line indicates an uniform acceleration of 10m/s2 for the green car.

The equation for velocity, v = u + at, is also the equation of a straight line where 'u' is the intial velocity, 'a' is the acceleration and 't' is the time instant. Therefore, in the velocity graph, the blue car traces a horizontal line at 30m/s as its acceleration is zero whereas the velocity of the green car increases linearly plotting a line with acceleration as the slope.

In real life, velocity and acceleration play an important role — for e.g. falling objects, boat in current, airplane in wind, trajectories of projectiles and lot more.

The Activity
The aim of the ‘Linear Motion’ exploriment is to enable the user to understand how the velocity changes with respect to acceleration and how it affects the time taken for a body to travel certain distance. The user can compare the motion of two cars by setting the initial velocity and acceleration values for each.

The motion of the two cars is plotted in terms of its velocity, acceleration and displacement v/s time. At the beginning the user can start off by keeping a constant acceleration of 0 m/s2 and observing the differences in time by altering the velocity. Gradually the acceleration can be altered and the time taken can be observed. This can also be done vice-versa (by keeping velocity constant and altering acceleration).

 

 

Formula & Numerical Problem

Equation for finding velocity of a body accelerating at a uniform rate

v = u + a.t

The equation for acceleration is

a = (v — u)/t

Distance travelled in time t by a body accelerating uniformly

s = u.t + ½at2

where,
u = initial velocity
t = time interval
v = velocity after time interval ‘t’
a = value of uniform acceleration
s = total distance travelled in time interval ‘t’

Numerical Example

A car accelerates from 15.0 m/s to 45.0 m/s in 10 seconds. Calculate the acceleration of the car. (Consider uniform acceleration). Calculate the distance travelled by the accelerating car during this period.

Given:
u (initial velocity) = 15.0 m/s.
v (final velocity) = 45.0 m/s
t (time) = 10 secs.
a (acceleration) = ?

The equation for acceleration:
a = (v — u)/t = (45.0 — 15.0)/10 = 30/10 = 3.0 m/s2

The acceleration of the car is given as 3.0 m/s2

For calculating the displacement,
s = ut + ½ at2

s = (30x10) + (3/2 x 102)

s = 450 m

The car will travel a distance of 450 meters during the given period of 10 seconds

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Theory | Formula
Instructions

This exploriment requires Adobe Shockwave plug-in. Download & install only if you do not see the exploriment on the left.

This exploriment looks at motion in a straight line and is mainly designed to help understand how acceleration affects the velocity of a moving body.

The activity includes two cars to help understand the concept by comparing the motion of two cars with different settings for entering velocity and acceleration. The cars leave trails to denote their path as they move,.which helps you to understand the motion of the cars between two points, A to B and also how the velocity changes due to acceleration.

The motion is also plotted on Displacement-Velocity-Acceleration v/s Time graphs.

Procedure
Set the entering (initial) velocities for the two cars.

Set the values for uniform acceleration, which you want each car to move with.

Click the 'Start' button to run the activity. You can abort the activity in between by clicking the 'Reset' button.

Observe the trails of the two cars to understand better the motion of the two cars between points A and B. Also note the graphs to see the effect of acceleration on the velocity.

Run the activity again at different settings.

Tips
The best way to understand the effect of acceleration is to run both cars at same entering velocity but different accelerations.

It is advisable to first set same velocity for both the cars and set zero acceleration (uniform velocity) for one car and some acceleration value for the second car. The difference in the trails of the two cars indicates quite clearly the effects of acceleration on motion.

Run the activity as many times as you want to with different inputs and try to draw your conclusions about the concept by observing the results. For more information, check out the 'Theory' section.