Motion Notes

Motion

In everyday life, we observe objects at rest and in motion, from birds flying to cars moving. Even at a microscopic level, atoms, molecules, planets, stars, and galaxies are in constant motion. We perceive motion when an object's position changes with time, although sometimes motion is inferred indirectly, like observing the movement of dust to infer the motion of air.

The phenomena of sunrise, sunset, and changing seasons are attributed to the Earth's motion, though we don't directly perceive it. An object's motion is relative; it may appear to be moving to one observer and stationary to another, such as trees appearing to move backwards for passengers on a moving bus. Motions can be complex, including straight-line, circular, rotational, and vibrational movements.

In this chapter, we will describe motion along a straight line using equations and graphs and discuss circular motion.

Describing Motion

To describe the location of an object, we specify a reference point, also called the origin. For example, a school 2 km north of the railway station uses the railway station as a reference point.

Motion Along a Straight Line

The simplest type of motion is motion along a straight line. Consider an object moving along a straight path, starting from point O (the reference point). Points A, B, and C represent the object's position at different times (Fig. 7.1).

The total path length covered by the object is OA + AC, which equals $60 km + 35 km = 95 km$ . This is the distance covered by the object which only requires a numerical value (magnitude) to be specified.

Displacement is the shortest distance from the initial to the final position. In the example where the object moves from O to A, the distance covered is $60 km$ , and the displacement is also $60 km$ . However, if the object moves from O to A and back to B, the distance covered is $60 km + 25 km = 85 km$ , while the displacement is $35 km$ . Therefore, the magnitude of displacement may be zero, while the corresponding distance covered is not zero. If the object returns to O, the displacement is zero, but the distance covered is $OA + AO = 60 km + 60 km = 120 km$ .

Distance and displacement are two different physical quantities used to describe motion. Distance requires only magnitude, while displacement requires both magnitude and direction.

Uniform Motion and Non-Uniform Motion

Uniform motion occurs when an object covers equal distances in equal intervals of time. For instance, if an object travels $5 m$ in the first second, $5 m$ in the next second, and so on, it is in uniform motion. Note: The time interval in this motion must be small.

Non-uniform motion occurs when objects cover unequal distances in equal intervals of time, such as a car moving on a crowded street.

Measuring the Rate of Motion

The rate at which objects move varies; some move fast, and some move slowly. One way to measure this rate is to find the distance travelled in unit time, which is called speed. The SI unit of speed is metre per second ( $m s^{-1}$ or $m/s$ ). Other units include centimetre per second ( $cm s^{-1}$ ) and kilometre per hour ( $km h^{-1}$ ).

The speed of an object need not be constant; in most cases, objects are in non-uniform motion. Therefore, we often describe the rate of motion in terms of average speed.

Average speed is calculated by dividing the total distance travelled by the total time taken:

$Average[space]speed = \frac{Total[space]distance[space]travelled}{Total[space]time[space]taken}$

If an object travels a distance $s$ in time $t$ , its speed $v$ is:

$v = \frac{s}{t}$ $(7.1)$

Speed With Direction

The rate of motion is more comprehensive when specifying direction along with speed. This quantity is called velocity. Velocity is the speed of an object moving in a definite direction. Like speed, velocity can be uniform or variable, and it can be changed by altering speed, direction, or both.

When an object moves along a straight line at a variable speed, we express the rate of motion in terms of average velocity:

$average[space]velocity = \frac{initial[space]velocity + final[space]velocity}{2}$

Mathematically, average velocity ( $v_{av}$ ) is:

$v_{av} = \frac{u + v}{2}$ $(7.2)$

where $u$ is the initial velocity and $v$ is the final velocity.

Speed and velocity have the same units: $m s^{-1}$ or $m/s$ .

Rate of Change of Velocity

During uniform motion along a straight line, velocity remains constant, and the change in velocity over any time interval is zero. However, in non-uniform motion, velocity varies with time. This change in velocity is described by acceleration, which is the measure of the change in velocity of an object per unit time.

$acceleration = \frac{change[space]in[space]velocity}{time[space]taken}$

If the velocity of an object changes from an initial value $u$ to a final value $v$ in time $t$ , the acceleration $a$ is:

$a = \frac{v - u}{t}$ $(7.3)$

This type of motion is called accelerated motion. Acceleration is positive if it is in the direction of velocity and negative if it is opposite to the direction of velocity. The SI unit of acceleration is $m s^{-2}$ .

If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time, the acceleration is uniform. The motion of a freely falling body is an example of uniformly accelerated motion.

On the other hand, if the velocity changes at a non-uniform rate, the object has non-uniform acceleration.

Graphical Representation of Motion

Graphs provide a convenient way to present information about various events. To describe the motion of an object, we use line graphs that show the dependence of physical quantities, such as distance or velocity, on time.

Distance-Time Graphs

The change in the position of an object with time is represented on a distance-time graph, with time taken along the x-axis and distance along the y-axis. These graphs can represent uniform speed, non-uniform speed, or objects at rest.

For uniform speed, the graph of distance travelled against time is a straight line, indicating that distance is directly proportional to time.

The speed ( $v$ ) of an object can be determined from the distance-time graph using the formula:

$v = \frac{s2 - s1}{t2 - t1}$ $(7.4)$

where $(s2 – s1)$ is the distance covered and $(t2 – t1)$ is the time interval.

For accelerated motion, the distance-time graph is a curve, showing non-linear variation of distance with time.

Velocity-Time Graphs

The variation in velocity with time is represented by a velocity-time graph, with time along the x-axis and velocity along the y-axis. If the object moves at uniform velocity, the graph is a straight line parallel to the x-axis.

The area enclosed by the velocity-time graph and the time axis equals the magnitude of the displacement.

For uniformly accelerated motion, the velocity-time graph is a straight line. The distance ( $s$ ) travelled by the object is given by:

$s = area[space]ABCDE = area[space]of[space]rectangle[space]ABCD + area[space]of[space]triangle[space]ADE$
$= AB × BC + \frac{1}{2} (AD × DE)$

In non-uniformly accelerated motion, velocity-time graphs can have any shape, representing the non-uniform variation of velocity with time.

Equations of Motion

When an object moves along a straight line with uniform acceleration, its velocity, acceleration, and the distance covered can be related through the equations of motion:

$v = u + at$ $(7.5)$
$s = ut + \frac{1}{2} at^2$ $(7.6)$
$2as = v^2 – u^2$ $(7.7)$

where:

$u$ is the initial velocity,
$v$ is the final velocity,
$a$ is the uniform acceleration,
$t$ is the time,
$s$ is the distance travelled.

Equation (7.5) describes the velocity-time relation, Equation (7.6) represents the position-time relation, and Equation (7.7) represents the relation between the position and the velocity.

Uniform Circular Motion

Acceleration occurs when the velocity of an object changes, either in magnitude or direction. Uniform circular motion is an example where an object does not change its speed but only its direction of motion. In this motion, an object moves in a circular path with constant speed.

The speed ( $v$ ) of an object moving in a circular path of radius $r$ is:

$v = \frac{2 \pi r}{t}$ $(7.8)$

where $t$ is the time taken to complete one revolution.