Velocity and Acceleration — Revision Notes

⚡ 30-Second Revision

Displacement (\( \vec{\Delta r} \)): — Change in position vector. Vector. Unit: m.
Average Velocity (\( \vec{v}_{avg} \)): — \( \frac{\Delta \vec{r}}{\Delta t} \). Vector. Unit: m/s.
Instantaneous Velocity (\( \vec{v} \)): — \( \frac{d\vec{r}}{dt} \). Vector. Unit: m/s.
Average Acceleration (\( \vec{a}_{avg} \)): — \( \frac{\Delta \vec{v}}{\Delta t} \). Vector. Unit: m/s\(^2\).
Instantaneous Acceleration (\( \vec{a} \)): — \( \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2} \). Vector. Unit: m/s\(^2\).
Graphical Analysis:

- Slope of x-t graph = velocity. - Slope of v-t graph = acceleration. - Area under v-t graph = displacement. - Area under a-t graph = change in velocity.

Key Concept: — Zero velocity does NOT imply zero acceleration (e.g., peak of projectile motion).

2-Minute Revision

Velocity and acceleration are the core descriptors of motion. Velocity is a vector, defining both speed and direction. Average velocity is total displacement over total time (\( \vec{v}_{avg} = \Delta \vec{r} / \Delta t \)), while instantaneous velocity is the velocity at a specific moment, found by differentiating position with respect to time (\( \vec{v} = d\vec{r}/dt \)).

Acceleration is also a vector, representing the rate of change of velocity. This change can be in speed, direction, or both. Average acceleration is total change in velocity over total time (\( \vec{a}_{avg} = \Delta \vec{v} / \Delta t \)), and instantaneous acceleration is found by differentiating velocity with respect to time (\( \vec{a} = d\vec{v}/dt \)) or twice differentiating position (\( \vec{a} = d^2\vec{r}/dt^2 \)).

Crucially, an object can have zero velocity but non-zero acceleration (like a ball at its highest point). Graphical analysis is vital: slopes of position-time, velocity-time graphs give velocity and acceleration respectively, while areas under velocity-time and acceleration-time graphs give displacement and change in velocity.

5-Minute Revision

To master velocity and acceleration for NEET, focus on their definitions, vector nature, and interrelationships, especially through calculus and graphical analysis. Velocity (\( \vec{v} \)) is the rate of change of position (\( \vec{r} \)), so \( \vec{v} = d\vec{r}/dt \).

It's a vector, meaning it has both magnitude (speed) and direction. If either changes, velocity changes. Acceleration (\( \vec{a} \)) is the rate of change of velocity, so \( \vec{a} = d\vec{v}/dt = d^2\vec{r}/dt^2 \).

It's also a vector. An object accelerates if it speeds up, slows down, or changes direction (even at constant speed, like in uniform circular motion).

Key Formulas:

Average Velocity: \( \vec{v}_{avg} = (\vec{r}_f - \vec{r}_i) / (t_f - t_i) \)
Average Acceleration: \( \vec{a}_{avg} = (\vec{v}_f - \vec{v}_i) / (t_f - t_i) \)

Graphical Interpretation:

Position-time (x-t) graph: — Slope gives instantaneous velocity. A straight line means constant velocity, a curve means changing velocity (acceleration).
Velocity-time (v-t) graph: — Slope gives instantaneous acceleration. A straight line means constant acceleration. Area under the curve gives displacement. Area above the axis is positive displacement, below is negative.
Acceleration-time (a-t) graph: — Area under the curve gives change in velocity.

Common Pitfalls:

1
Speed vs. Velocity: — Don't confuse them. Speed is scalar, velocity is vector. Constant speed with changing direction means changing velocity and thus acceleration.
2
Zero Velocity, Non-Zero Acceleration: — An object can momentarily stop (zero velocity) but still be accelerating (e.g., a ball at the peak of its throw, where \( a = g \) downwards).
3
Negative Acceleration: — Doesn't always mean slowing down. If velocity is negative and acceleration is negative, the object is speeding up in the negative direction.

Example: If \( x(t) = 5t^2 - 3t \), then \( v(t) = dx/dt = 10t - 3 \) and \( a(t) = dv/dt = 10 \). This shows constant acceleration.

Prelims Revision Notes

For NEET, a solid understanding of velocity and acceleration is non-negotiable. Remember that velocity is a vector quantity, encompassing both speed and direction. Average velocity is calculated as total displacement divided by total time (\( \vec{v}_{avg} = \Delta \vec{r} / \Delta t \)).

Instantaneous velocity is the velocity at a specific moment, obtained by differentiating the position function with respect to time (\( \vec{v} = d\vec{r}/dt \)). Its magnitude is instantaneous speed.

Acceleration is also a vector, representing the rate of change of velocity. This change can be in magnitude (speeding up or slowing down) or direction. Average acceleration is the total change in velocity divided by total time (\( \vec{a}_{avg} = \Delta \vec{v} / \Delta t \)).

Instantaneous acceleration is the acceleration at a specific moment, found by differentiating the velocity function with respect to time (\( \vec{a} = d\vec{v}/dt \)) or by taking the second derivative of the position function (\( \vec{a} = d^2\vec{r}/dt^2 \)).

Key relationships to recall:

If velocity and acceleration are in the same direction, the object speeds up.
If velocity and acceleration are in opposite directions, the object slows down.
An object moving with constant velocity has zero acceleration.
An object can have zero velocity at an instant but still have non-zero acceleration (e.g., a ball thrown upwards at its peak, where \( v=0 \) but \( a=g \) downwards).

Graphical Analysis is crucial:

Position-time (x-t) graph: — The slope gives velocity. A horizontal line means rest. A straight, inclined line means constant velocity. A curve means changing velocity (acceleration).
Velocity-time (v-t) graph: — The slope gives acceleration. A horizontal line means constant velocity (zero acceleration). A straight, inclined line means constant acceleration. The area under the curve gives displacement. Area above the x-axis is positive displacement, below is negative.
Acceleration-time (a-t) graph: — The area under the curve gives the change in velocity.

Practice problems involving differentiation and integration for polynomial functions of time, and extensively practice interpreting all three types of motion graphs.

Vyyuha Quick Recall

VAD: Velocity is Acceleration's Derivative. (And position's derivative is velocity!)