In this article, the topic is the relationship between **Newton's Second Law and Motion,** and the **displacement **of an object.

The method here will demonstrate how to find the force needed to move an object.

*How do you*** find the force needed** given time (t), final displacement (df), initial displacement (di), initial velocity (vi), and the mass of the object?

Once again, it is necessary to combine two principles in order to see the full relationship.

You will have to find the rate of acceleration (a) using the distance equation, then use that to find the force (F).

So, you want to use

*F=ma*

to find F, but you don't have the a. That means finding a using the distance information given and the distance equation.

The distance formula is:

*df = ** di +** (vi )(t)+ 1/2(a)(t*^{2})

**So, we'll be doing two steps.**

Step 1:

Use the distance equation to find a:

*df = ** di +** (vi )(t)+ 1/2(a)(t*^{2})

Step 2:

Then use the calculated a with Newton's Second Law

*F = ma*

to find the force.

EXAMPLES

Example 1

Let's take a look at another example (even easier!) and work it out:

**What force acts on a object** with a mass of 10 kg, if it begins 5 meters from a mark and has an initial velocity of 25 m/s, and ends up a final distance of 227 m from the mark after an elapsed time of 6 seconds?

STEP 1

Find a where:

df = 227 m

di = 5 m

vi = 25 m/s

t = 6 s

*df = ** di +** (vi )(t)+ 1/2(a)(t*^{2})

227 m = 5 m + (25 m/s)(6 s) + 1/2*(a)(6*^{2})

*Combine some terms... and PEMDAS*

*227 m = 5 m + (25 m/s)(6 s) + 1/2**(a)(6 s)*^{2}

227 m = 5 m + 150 m + 1/2(a)(36 s^{2})

227 m = 155 m + (18s^{2})(a)

Subtract 155 m from both sides...

78 m = (18s^{2})(a)

Divide both sides by 18s^{2} ...

78 m / 18s^{2} = a

*4 m/s/s = a*

Next, use THAT calculated a to find the Force (step 2 above):

Find F where

m = 10 kg

a = 4 m/s/s

*F = ma*

*F = (10 kg )(4 m/s/s)*

*F = 40 N*

How about *seeing one* worked out?

**NOTE**

While this example shows how to find F when given distance information, the same process applies when asked to find the force given velocity information. However, you'd begin (Step 1) using the velocity information and formula to find a, the go on to Step 2 and solve for force.

SUMMARY:

You gotta do it in steps!

This process requires doing the work in steps. Depending on what is given, you use the two formulas below:

*F = ma*

*df = ** di +** (vi )(t)+ 1/2(a)(t*^{2})

So, read the problem, write down what is being looked for, write down what is given, THEN... use the two formulas. Two steps!