🚗 Average Acceleration 🚨Instantaneous Acceleration(Motion in two dimension | Class 11 Physics

In this video, we will explore the concept of instantaneous acceleration of a particle in two dimensions. We will cover topics such as the acceleration formula, average acceleration, instantaneous acceleration, vector acceleration, and their implications.

Acceleration is a measure of how the velocity of a particle changes. If the velocity of a particle changes, it is said to be accelerating. Even if the magnitude of velocity remains constant, but the direction of motion changes, the body is still considered to be accelerating. For example, if a rocket moving through space at 8000 m/s changes its trajectory and starts moving at 8000 m/s in a curve, we can say that the rocket is accelerating.

Let's start by understanding the idea of average acceleration. When a particle's velocity changes from v₁ to v₂ over a time interval Δt, its average acceleration in that time interval is given by the formula:
Average acceleration = (v₂ - v₁) / Δt

To find the average acceleration between two positions P₁ and P₂, we need to find the change in velocity (Δv), which is equal to v₂ - v₁. Since both v₂ and v₁ are vectors, we perform a vector operation to find Δv. Dividing Δv by Δt gives us the average acceleration between P₁ and P₂. This average acceleration should be in the direction of the change in velocity or vector Δv, as we are dividing Δv by a scalar.

Now, let's delve into the concept of instantaneous acceleration. If we shrink Δt towards zero, continuously reducing it to very small values like 0.1s, 0.001s, 0.0001s, and so on, the ratio Δv / Δt becomes the instantaneous acceleration at point P₁. As we reduce the time interval, we approach point P₁, and any value of Δv / Δt gives us a good approximation of the acceleration at that point. This can be written as the limit of Δt tending to zero, expressed as α = Δv / Δt.

An interesting way to analyze instantaneous acceleration is by examining its components separately. When a particle is subjected to acceleration acting in a certain direction, the acceleration vector can be divided into two components: one parallel to the velocity vector and another perpendicular to the velocity vector.

The parallel component of acceleration tells us about the change in the particle's speed, while the perpendicular component indicates a change in the particle's direction of motion. The perpendicular component "tugs" the particle in its direction, while the particle wants to move in the direction parallel to the velocity vector. As a result, a curved trajectory is created.

Now, let's consider two situations. In the first situation, the acceleration vector is in the same direction as the velocity vector v₁. In this case, the acceleration has a parallel component only, and the perpendicular component is zero. Consequently, the change in velocity would also be in the same direction as the acceleration. The velocity v₂ after time Δt would also be in the same direction as v₁, resembling motion in a straight line.

In the second situation, the acceleration is perpendicular to the velocity vector. Consequently, the acceleration has no component parallel to the velocity vector. During a time interval Δt, the change in velocity (Δv) will happen in the direction of the acceleration. To find the final velocity v₂, we add Δv to the initial velocity vector v₁. As we decrease Δt, Δv becomes smaller. If we significantly reduce Δt, we find that the magnitude of v₂.

Key Moments
0:00: Introduction to the concept of acceleration
0:22: Example of a rocket changing velocity and accelerating
0:46: Introduction to average and instantaneous acceleration
1:00: Formula for average acceleration: ΔV/ΔT
1:09: Example of finding average acceleration between two positions
2:16: Average acceleration as a vector in the direction of ΔV
2:59: Introduction to instantaneous acceleration
3:06: Definition of instantaneous acceleration as ΔV/ΔT when ΔT approaches 0
4:11: Instantaneous acceleration notation: a = dV/dt or a = (dvx/dt)i + (dvy/dt)j + (dvz/dt)k
5:20: Explanation of acceleration components and their effects on speed and direction
6:25: Analysis of acceleration components in different scenarios
8:08: Effect of acceleration vector direction on velocity change

LESSON SUMMARY:
1. Acceleration occurs whenever the velocity of a particle changes, including changes in direction.
2. Average acceleration is determined by the change in velocity divided by the time interval: ΔV/ΔT.
3. Instantaneous acceleration is the limit of average acceleration as the time interval approaches zero: a = dV/dt.
4. Instantaneous acceleration can be represented using unit vectors: a = (dvx/dt)i + (dvy/dt)j + (dvz/dt)k.
5. The parallel component of acceleration affects the speed of the particle.
6. The perpendicular component of acceleration affects the direction of the particle's motion.