# Find the instantaneous velocity

• pbonnie
In summary, the instantaneous velocity of a ball thrown in the air at t=2 seconds can be found by taking the derivative of its height function, h(t) = -5t^2 + 20t + 1, which is -10t + 20. Plugging in t=2, the instantaneous velocity is 0, indicating a change in direction.
pbonnie

## Homework Statement

a ball is thrown in the air. its height from the ground in metres after t seconds is modeled by h(t) = -5t^2 + 20t + 1. What is the instantaneous velocity of the ball at t=2 seconds?

(f(a+h)-f(a))/h

## The Attempt at a Solution

h(2) = -5(2)2 + 20(2) + 1 = 21
h(2+h) = -5(2+h)2 + 20(2+h) + 1
= -5(2+h)(2+h) + 40 + 20h + 1
= -5(4+2h+2h+h2) + 20h + 41
=-20 -10h – 10h-5h2 + 20h + 41
= -5h2 + 21
lim┬(h→0)⁡〖(f(2+h)-f(2))/h〗 = ((-5h^2+ 21)- 21)/h = (-5h^2)/h = -5h
But then I'm supposed to let h = 0... so the instantaneous velocity is 0. This seems wrong to me? Unless because this is the maximum value, that is when the direction changes, so the velocity is 0?
Thank you for helping!

pbonnie said:

## Homework Statement

a ball is thrown in the air. its height from the ground in metres after t seconds is modeled by h(t) = -5t^2 + 20t + 1. What is the instantaneous velocity of the ball at t=2 seconds?

(f(a+h)-f(a))/h

## The Attempt at a Solution

h(2) = -5(2)2 + 20(2) + 1 = 21
h(2+h) = -5(2+h)2 + 20(2+h) + 1
= -5(2+h)(2+h) + 40 + 20h + 1
= -5(4+2h+2h+h2) + 20h + 41
=-20 -10h – 10h-5h2 + 20h + 41
= -5h2 + 21
lim┬(h→0)⁡〖(f(2+h)-f(2))/h〗 = ((-5h^2+ 21)- 21)/h = (-5h^2)/h = -5h
But then I'm supposed to let h = 0... so the instantaneous velocity is 0. This seems wrong to me? Unless because this is the maximum value, that is when the direction changes, so the velocity is 0?
Thank you for helping!

Yes, it is the maximum value and that's where the direction changes. 0 is correct.

Great, thank you :)

## 1. What is instantaneous velocity?

Instantaneous velocity is the rate of change of an object's position at a specific moment in time. It is calculated by taking the derivative of the object's position function with respect to time.

## 2. How is instantaneous velocity different from average velocity?

Average velocity is the total displacement of an object divided by the total time taken, while instantaneous velocity is the velocity at a specific point in time. Average velocity gives an overall picture of an object's motion, while instantaneous velocity gives information about the object's motion at a specific moment.

## 3. Can instantaneous velocity be negative?

Yes, instantaneous velocity can be negative. A negative velocity indicates that the object is moving in the opposite direction of its positive motion.

## 4. How is instantaneous velocity measured?

Instantaneous velocity can be measured using a variety of methods, such as using a speedometer, radar gun, or a motion sensor. The object's position and time data can also be collected and used to calculate the instantaneous velocity using the derivative formula.

## 5. Why is instantaneous velocity important in science?

Instantaneous velocity is important in science because it helps us understand the precise motion of objects. It allows us to calculate the acceleration of an object, which is essential in studying the laws of motion and predicting future motion. Instantaneous velocity is also used in various fields such as physics, engineering, and astronomy to analyze and model complex motion.

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