Newton's Second Law and Displacement

In this article, the topic is the relationship between Newton's Second Law and Motion, and this time, displacement will be the focus.

How do you find a total displacement when given time (t), initial displacement (di), initial velocity (vi), but instead of acceleration (a), you are given force (F) and mass (m)?

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

The distance formula is:

df =  di  +  (vi )(t)+ 1/2(a)(t²)

However, to find final displacement, you need an acceleration(a), but you have mass and force.

Using Newton's Second Law, fortunately, will allow the calculation of a using the force and mass given.

So, we'll be doing two steps.

Step 1: 

Use 

F=ma to find a

Step 2:

Then use THAT in the distance formula.

df =  di  +  (vi )(t)+ 1/2(a)(t²)

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EXAMPLES

Example 1

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

A force of 50 N acts on a object with a mass of 10 kg. If it has an initial displacement (di) of 10 m and has an initial velocity (vi) of 20 m/s, what is its final displacement after an elapsed time of 4 seconds?

The question tells us that we need to find final velocity (vf). But there is no acceleration given. So, do step 1 (above):

Find a where:

F = 50 N

m = 10 kg

F = ma

50 N = (10 kg)(a)

50 N/10 kg = a

5 m/s/s = a

Next, use THAT calculated a to find final velocity (step 2 above):

Find df where

di = 10 m

vi = 20 m/s

t = 4 s

a = 5 m/s/s

df =  di  +  (vi )(t)+ 1/2 (a)(t²)

df = 10 m  +  (20 m/s )(4 s)+ 1/2 (5 m/s/s)(4²)

df = 10 m  +  80 m + 1/2 (5 m/s/s)(16 s²)

df = 10 m  +  80 m + 40 m
df = 130 m

Example 2

How about seeing one worked out?

https://youtu.be/TjxUEwMjmvc

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)(t²)

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

Newton's Second Law and Final Velocity

Suppose you are faced with a problem such as:

A motorcycle stunt rider needs to reach a final velocity of 799 m/s in order to make the jump needed for the scene in the movie. If she has an initial velocity of 10 m/s and the total mass of her and the motorcycle is 125 kg and if she can create a force of 100 N, how much time will be needed to achieve the needed final velocity?

On the surface, it looks (exciting, but also) like this could be hard to do.

IT'S NOT!

The velocity formula is very easy:

vf = vi + (a)(t)

In the problem above, you are looking for t and vf and vi are given. And then there's that mass and force...

You need an acceleration(a), but you have mass and force. Thankfully Newton did that thing:

F = ma

So, we'll be doing two steps.

Step 1: 

Use 

F=ma to find a

Step 2:

Then use THAT in the velocity formula.

vf = vi + (a)(t)

---

EXAMPLES

Example 1

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

A force of 50 N acts on a object with a mass of 10 kg. If it has an initial velocity of 20 m/s, what is its final velocity after an elapsed time of 8 seconds?

The question tells us that we need to find final velocity (vf). But there is no acceleration given. So, do step 1 (above):

Find a where:

F = 50 N

m = 10 kg

F = ma

50 N = (10 kg)(a)

50 N/10 kg = a

5 m/s/s = a

Next, use THAT calculated a to find final velocity (step 2 above):

Find vf where

vi = 20 m/s

t = 8 s

a = 5 m/s/s

vf = vi + (a)(t)

vf = 20 m/s + (5 m/s/s)(8 s)

vf = 20 m/s + 40 m/s

vf = 60 m/s

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Example 2

How about seeing it worked out?

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

vf = vi + (a)(t)

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