Accelerated Motion

$$
\fbox{acceleration}
\cdots velocity\ change\ per\ unit\ time
$$

How fast do you run?
How fast is the Shinkansen?
The Shinkansen moves very fast. So you can say "The Shinkansen has a high speed" or "Shinkansen has large velocity."
But if you and the Shinkansen start moving at the same time, who would reach 5 m/s first? In other words,  who would win in a 50 meter sprint?
Maybe you.
At that point, you can say "You had a greater acceleration than the Shinkansen."
Acceleration represents the change in speed, or more technically, change in velocity per unit time.

Assuming Taro’s velocity changed from 5 m/s to 8 m/s in 2 seconds, his acceleration was 1.5 m/s² because it changed by 3 m/s in 2 seconds. The calculation is shown below:

$$
acceleration = \frac{8(m/s)-5(m/s)}{2(s)}\\
a = \frac{3(m/s)}{2(s)}\\
a = 1.5 (m/s^2)
$$

We usually use v₀ as initial velocity when observer starts observing. The stopwatch represents "zero" at that time, so we denote the initial velocity as v₀ (means velocity_0).
The velocity at any given time or the final velocity is simply written as v. More technically, we denote it as v(t) because v is a dependent variable of t. (meaning v as a function of t )

$$
acceleration=\frac{final\ velocity - initial\ velocity}{time\ span}=\frac{velocity\ change}{time\ span}\\
a=\frac{v - v_0}{\Delta t}=\frac{\Delta v}{\Delta t}\\
$$

You don't have to use "time span". You can just use "time" represented by the stopwatch. So, the equation below shows almost the same meaning.

$$
a=\frac{v - v_0}{t}=\frac{\Delta v}{t}
$$

Let's practice.

Ex1: A stationary Hanako starts moving and after 2 seconds, she reaches a speed of 6 m/s. What is Hanako's acceleration at this time?

Ex2: Ken moving at 5 m/s undergoes uniform acceleration and after 3 seconds, he reaches a velocity of -1 m/s. What is Ken's acceleration at this time?

Ans1: 3 m/s².

$$
accel=\frac{6-0\ (m/s)}{2\ (s)}=3\ (m/s^2)
$$

Ans2: -2 m/s².

$$
a=\frac{-1-5\ (m/s)}{3\ (s)}=-2\ (m/s^2)
$$

Actually, we have to distinguish between "speed" and "velocity", as one is a scalar and the other is a vector.
But that will be a topic for next time.

illust : Copilot