MOTION IN A STRAIGHT LINE-Notes

Instantaneous velocity is the velocity of a body at a particular instant of time. It describes both how fast the object is moving and in which direction, at that exact moment.

Suppose an object moves along a straight line and its position changes with time. Over a small time interval \(\Delta t\), its displacement is \(\Delta x\). The average velocity is

\[v_{avg}=\dfrac{\Delta x}{\Delta t}\]

Now imagine making the time interval smaller and smaller. As \(\Delta t\) becomes extremely tiny, the average velocity over that interval approaches the velocity at that instant. This limiting value is the instantaneous velocity.
Mathematically,

\[v=\lim_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t}=\dfrac{dx}{dt}\]

So, instantaneous velocity is the rate of change of position with respect to time.

Important features

• It can be positive, negative, or zero, depending on direction.
• It depends on the direction of motion along the straight line.
• Even when speed changes continuously, instantaneous velocity is well-defined at every instant.

Acceleration is defined as the rate of change of velocity with respect to time.

If a body’s velocity changes from \(u\) to \(v\) in a time interval \(\Delta t\), the average acceleration is

\[a_{avg}=\dfrac{v-u}{\Delta t}\]

When we consider an extremely small time interval, the acceleration at a particular instant is called instantaneous acceleration:

\[a=\lim_{\Delta t \to 0}\dfrac{\Delta v}{\Delta t}=\dfrac{dv}{dt}\]

Thus, acceleration tells us how rapidly and in what manner velocity is changing at a given moment.

Nature of Acceleration

• Acceleration is a vector quantity.
• Its direction is the same as the direction of change in velocity, not necessarily the direction of motion.
• In straight-line motion, acceleration can be:
  • positive (velocity increasing in the chosen positive direction),
  • negative (velocity decreasing or increasing in the opposite direction),
  • zero (velocity constant).

Uniform and Non-Uniform Acceleration

Uniform acceleration

A body has uniform acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the intervals are.

Example: A body moving under gravity near the Earth’s surface (ignoring air resistance).

Non-uniform acceleration

A body has non-uniform acceleration if the change in velocity is unequal in equal time intervals.

Example: A car moving through traffic, repeatedly speeding up and slowing down.

Special Cases of Acceleration

• Zero acceleration:
  When velocity remains constant (both magnitude and direction), acceleration is zero. The body may still be moving, but its motion is uniform.
• Negative acceleration (Retardation):
  When acceleration acts opposite to the direction of velocity, the speed of the body decreases. This is often called retardation or deceleration.
• Acceleration with constant speed:
  Even if speed remains constant; acceleration can exist if velocity changes direction. (This idea becomes clearer in circular motion, though direction change is limited in straight-line motion.)

Unit and Dimensions of Acceleration

• SI unit: metre per second squared \[(m\ s^{−2})\]
• Dimensions:\[[LT^{-2}]\]

These equations can also be obtained by Method of Calculus

By Definition \[\begin{aligned} a&=\dfrac{dv}{dt}\\ dv&=adt \end{aligned}\] Integrating both sides \[\begin{aligned} \displaystyle \int_{v_0}^{v} dv &= \int_{0}^{t} a\,dt\\\\ \displaystyle \int_{v_0}^{v} dv &= a\,\int_{0}^{t} dt \quad\scriptsize (a\text{ is constant})\\\\ v-v_0&=at\\ \Rightarrow v&=v_0+at \end{aligned}\] \[\boxed{\;\boldsymbol{v=v_0+at}\;}\] Further \[\begin{aligned}v&=\dfrac{dx}{dt}\\\\ dx&=v\,dt\end{aligned}\] Integrating both sides \[\begin{aligned} \displaystyle\int_{x_0}^x dx&=\int_{0}^{t} v\,dt\\\\ \displaystyle\int_{x_0}^x dx&=\int_{0}^{t} (v_0+at)dt\\\\ &=\int_{0}^{t} (v_0+at)dt\\\\ x-x_0&=v_0t+\dfrac{1}{2}at^2 \end{aligned}\] \[\boxed{\;\boldsymbol{x=x_0+v_0\,t+\dfrac{1}{2}at^2}\;}\] We can write \[ \begin{aligned} a&=\dfrac{dv}{dt}\\\ &=\dfrac{dv}{dx}\dfrac{dx}{dt}\\\\ &=\dfrac{dv}{dx}\dfrac{dx}{dt}\\\\ &=v\dfrac{dv}{dx}\\\\ a\,dx=v\,dv \end{aligned} \] Integrating both sides \[ \begin{aligned} \int_{x_0}^x a\,dx&= \int_{v_0}^v v\,dv\\\\ a\,(x-x_0)&=\dfrac{1}{2}(v^2-v_0^2)\\\\ 2a\,(x-x_0) &= (v^2-v_0^2)\\\\ \Rightarrow (v^2-v_0^2) &=2a\,(x-x_0) \end{aligned} \] or \[\boxed{\;\boldsymbol{v^2=v_0^2+2a\,(x-x_0)}\;}\]

The advantage of this method is that it can be used for motion with non-uniform acceleration also.

An object is said to be in free fall when it moves only under the influence of gravity, with no other forces acting on it. When air resistance is neglected, gravity becomes the sole cause of motion.

Examples include:

• A stone dropped from a height
• A ball thrown vertically upward or downward (after leaving the hand)

In all such cases, the motion is governed entirely by Earth’s gravitational pull.

Acceleration in Free Fall

Near the Earth’s surface, every freely falling object experiences a constant downward acceleration, called the acceleration due to gravity, denoted by \(g\).

• Magnitude of \(g\approx 9.8\ m\ s^{-2}\)
• Direction: vertically downward
• Same for all objects, regardless of their mass or shape (when air resistance is ignored)

This means a heavy stone and a light feather would fall together if air resistance were absent.

Nature of Motion

Free fall is a case of uniformly accelerated motion along a straight vertical line.

• Acceleration remains constant.
• Velocity changes uniformly with time.
• Motion can be either downward (object dropped or thrown downward) or upward (object thrown vertically upward).

Equations of Motion in Free Fall

The standard kinematic equations apply, with acceleration \(a=g\) (or \(−g\), depending on the chosen sign convention).

If the downward direction is taken as positive:

1. Velocity–time relation \[v=u+gt\]
2. Displacement–time relation> \[s=ut+gt^2\]
3. Velocity–displacement relation \[v^2=u^2+2gs\]

Galileo’s law of odd numbers states that:

For a body starting from rest and moving with uniform acceleration, the distances covered in successive equal intervals of time are in the ratio of consecutive odd natural numbers:

1:3:5:7:…

This simple law gives a deep insight into how distance grows when velocity increases steadily with time.

Important conditions

Galileo’s law holds only when:

• The body starts from rest
• Motion is along a straight line
• Acceleration is constant
• Time intervals are equal
If any of these conditions change, the odd-number pattern no longer applies.

The stopping distance of a vehicle is the total distance travelled by the vehicle from the moment the driver decides to stop until the vehicle comes to a complete rest.

It is not just the distance covered after applying the brakes. It has two distinct parts, both equally important in real-life motion.

Components of Stopping Distance

Stopping distance=Reaction distance+Braking distance

Reaction Distance

Reaction distance is the distance travelled by the vehicle during the driver’s reaction time—that short interval between seeing a hazard and actually applying the brakes.

Braking Distance

Braking distance is the distance the vehicle travels after the brakes are applied, until it stops completely.

When the brakes are applied, the vehicle undergoes uniform retardation due to the friction between the tyres and the road.

Total Stopping Distance

\[S_{total}=vt_r + \dfrac{v^2}{2a}\]

Where,
\(vt_r\) is Reaction distance
\(a\) is retardation due to applying Brakes
\(v\) is the velocity

Reaction time is the time interval between the moment a person perceives a stimulus and the moment they respond to it. In the context of motion, especially road safety, it refers to the time taken by a driver to see a hazard and begin an action, such as pressing the brake pedal.

During this short interval, the vehicle continues to move with its original speed, even though the driver intends to stop.