$$ s''(t)=v'(t)=a(t) $$

Position, Velocity and Acceleration

• s(t): the function with the input of t and output of the position of the object.
• v(t): the function with the input of t and output of the velocity of the object.
• a(t): the function with the input of t and output of the acceleration of the object.
  • t=time

Speed, Velocity and Acceleration

<aside> 💡 Things to review

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Speed: a scalar quantity, where only the “size” is available. THE MAGNITUDE DOES NOT EXIST. Therefore, the speed of an object is the absolute value of the change in y over change in x, by the application that is the position function that we need. In another word, the absolute value of the Velocity.

• If the question ask you the speed at one instance t of the function, it is asking for the absolute value of the derivative of the position function, input of t - time.

Velocity: a vector quantity, where both the magnitude and the “size” is available. Therefore, there is negative and positive of the function’s output. The definition of velocity is the difference of position over the difference in time.

• If the question ask you the velocity at one instance t of the function, it is asking for the derivative of the position function, input of t - time.

Acceleration: a vector quantity, where both the magnitude and the “size” is available. Therefore, there is negative and positive of the function’s output. The definition of acceleration is the difference of velocity over the difference in time. (Therefore, derivative of v in respect of t)

• If the question ask you the acceleration at one instance t of the function, it is asking for the derivative of the velocity function, input of t - time.

Speeding Up, Constant ,or Down?

Since we know the definition of speed, lets demonstrate this visually by car on the road.

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s 0 +

1. If the velocity is negative, the car will travel west (negative numbers)…
   1. Speeding up: Therefore, the acceleration at the current location must be negative; to make the car travel west faster
   2. Slowing down: Therefore, the acceleration at the current location must be positive; to make the car travel slower, began to head east.
   3. Constant: The acceleration must be zero, for the velocity have no changes.
2. If the velocity is positive, the car will travel east (positive numbers), therefore, the acceleration at the current location must be negative; to make the car travel west faster
   1. Speeding up: Therefore, the acceleration at the current location must be positive; to make the car travel east faster
   2. Slowing down: Therefore, the acceleration at the current location must be negative; to make the car travel slower, began to head west.
   3. Constant: The acceleration must be zero, for the velocity have no changes.