# Deriving Motion Equation from Newton's Second Law

## Homework Statement

An object is dropped from a height above a pool of water.

Start with Newton’s 2nd law, F = ma and use the general expression for acceleration, ay=d2y/dt2 to generate a differential equation

F=ma

-mg=m d2y/dt2

## The Attempt at a Solution

We had this problem in our problem solving session last week. My physics I class I never learned how to derive anything, so I am always on a quest to figure these out. This is one I cannot do. I can set up as far as I have. I see I can cancel the m's and then integrate both sides, but I cannot get to the correct answer.

The answer is a regular equation of motion - y1=y2+vt - 1/2gt2.

How does one approach this problem?

LCKurtz
Homework Helper
Gold Member

## Homework Statement

An object is dropped from a height above a pool of water.

Start with Newton’s 2nd law, F = ma and use the general expression for acceleration, ay=d2y/dt2 to generate a differential equation

F=ma

-mg=m d2y/dt2

## The Attempt at a Solution

We had this problem in our problem solving session last week. My physics I class I never learned how to derive anything, so I am always on a quest to figure these out. This is one I cannot do. I can set up as far as I have. I see I can cancel the m's and then integrate both sides, but I cannot get to the correct answer.

The answer is a regular equation of motion - y1=y2+vt - 1/2gt2.

How does one approach this problem?

You have the right equation to start.

y''(t) = -g

Here's the next step. Integrate once:

y'(t) = -gt + C

Use the velocity at t = 0 to figure out C, then integrate again, getting another constant you will need to figure out from the initial conditions.