# Integration and initial velocity

• toasted
In summary, the problem is asking for the initial velocity needed for an object to reach a maximum height of 550 feet when thrown upward from ground level, neglecting air resistance and using the acceleration due to gravity of -32 ft/s2. The solution involves integrating the given acceleration to find the velocity as a function of time, where the integration constant can be determined by considering the highest point in the velocity graph or formula.
toasted

## Homework Statement

With what initial velocity must an object be thrown upward (from ground level) to reach a maximum height of 550feet.

Use a(t)= -32ft/sec2 as the acceleration due to gravity. (neglect air resistance)

Use integration

## The Attempt at a Solution

I know that U should first start off by integrating the acceleration in order to get velocity, but I wind up getting:

32x+C= v(t)

I'm not sure how to deal wit the problem from here, does anyone have any suggestions?

What is x?

Im pretty sure that x is suppose to be time, because that is the only thing that relates acceleration and velocity

I thought t was time as well. So lesson one is to be a bit more precise in your notation.

a(t) = -32 ft/s2, then
$$v(t) = \int a(t) \, dt$$
where t is the variable. Integrating a constant over t gives you the constant times t so
v(t)[ft/s] = - 32 t + C

Note the minus sign, which is carried over from a(t) < 0.
C is an integration constant which you need to determine. What condition will you use for this?
How can you see in the v(t) graph or formula that the highest point is reached?

## What is integration?

Integration is a mathematical process that involves finding the area under a curve. It is used to solve problems involving accumulation, such as finding the total distance traveled or the total amount of a substance produced over time.

## How is integration related to initial velocity?

Initial velocity is the velocity of an object at the beginning of its motion. Integration can be used to find the displacement of an object by integrating its initial velocity over time.

## What is the difference between integration and differentiation?

Differentiation is the inverse of integration and involves finding the rate of change of a function at a specific point. Integration, on the other hand, finds the accumulated amount of a function over a given interval.

## How is initial velocity calculated?

Initial velocity can be calculated by dividing the change in displacement by the change in time. It can also be found by taking the derivative of the displacement function at time t=0.

## What are some real-world applications of integration and initial velocity?

Integration and initial velocity are used in many fields, including physics, engineering, and economics. Some examples of real-world applications include calculating the distance traveled by a car, finding the amount of medication in a patient's bloodstream, and determining the rate of change of stock prices over time.

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