# Newtonian mechanics

## Introduction

### SUVAT

#### Introduction

For a constant acceleration environment we want to find equations to link:

• Initial speed: $$v_{t_0}$$

• End speed: $$v_{t_1}$$

• Time: $$t_1-t_0$$

• Acceleration: $$a$$

• Displacement $$s_{t_1}-s_{t_0}$$

#### The SUVAT equations

These are the following, and are derived below.

• $$v_{t_1}=a(t_1-t_0)+v_{t_0}$$

• $$(s_{t_1}-s_{t_0})=v_{t_0}(t_1-t_0)+\dfrac{1}{2}a(t_1-t_0)^2$$

• $$(s_{t_1}-s_{t_0})=v_{t_1}(t_1-t_0)-\dfrac{1}{2}a(t_1-t_0)^2$$

• $$v_{t_1}^2= v_{t_0}^2+2a(s_{t_1}-s_{t_0})$$

• $$(s_{t_1}-s_{t_0})=(t_1-t_0)\dfrac{v_{t_1}+v_{t_1}}{2}$$

#### Equation 1: No displacement

This equation is:

$$v_{t_1}=a(t_1-t_0)+v_{t_0}$$

$$v_t:=\dfrac{\delta s_t}{\delta t}$$

$$a:=\dfrac{\delta v_t}{\delta t}$$

If acceleration is constant, then

$$\dfrac{\delta v_t}{\delta t}=a$$

$$v_t=\int a dt +v_0$$

$$v_t=at+v_0$$

#### Equation 2: No end velocity

This equation is:

$$(s_{t_1}-s_{t_0})=v_{t_0}(t_1-t_0)+\dfrac{1}{2}a(t_1-t_0)^2$$

$$v:=\dfrac{\delta s_t}{\delta t}$$

Then:

$$\dfrac{\delta s_t}{\delta t}=at+v_0$$

$$s_t=\dfrac{1}{2}at^2+v_0t+s_0$$

$$(s_t-s_0)=v_0t+\dfrac{1}{2}at^2$$

#### Equation 3: No start velocity

This equation is:

$$(s_{t_1}-s_{t_0})=v_{t_1}(t_1-t_0)-\dfrac{1}{2}a(t_1-t_0)^2$$

$$v_t=at+v_0$$

$$(s_t-s_0)=t\dfrac{v_t+v_0}{2}$$

So:

$$v_0=v_t-at$$

$$v_0=\dfrac{2}{t}(s_t-s_0)- v_t$$

$$v_t-at=\dfrac{2}{t}(s_t-s_0)- v_t$$

$$(s_t-s_0)=v_tt-\dfrac{1}{2}at^2$$

#### Equation 4: No time

This equation is:

$$v_{t_1}^2= v_{t_0}^2+2a(s_{t_1}-s_{t_0})$$

$$v_t=at+v_0$$

$$(s_t-s_0)=t\dfrac{v_t+v_0}{2}$$

So:

$$t=\dfrac{v_t-v_0}{a}$$

$$t=2\dfrac{s_t-s_0}{v_t+v_0}$$

$$\dfrac{v_t-v_0}{a}=2\dfrac{s_t-s_0}{v_t+v_0}$$

$$(v_t-v_0)(v_t+v_0)=2a(s_t-s_0)$$

$$v^2_t= v^2_0+2a(s_t-s_0)$$

#### Equation 5: No acceleration

This equation is:

$$(s_{t_1}-s_{t_0})=(t_1-t_0)\dfrac{v_{t_1}+v_{t_1}}{2}$$

$$v_t=at+v_0$$

$$s_t-s_0=\dfrac{1}{2}at^2+v_0t$$

So:

$$a=\dfrac{v_t-v_0}{t}$$

$$a=\dfrac{2[(s_t-s_0)-v_0t]}{t^2}$$

$$\dfrac{v_t-v_0}{t}=\dfrac{2[(s_t-s_0)-v_0t]}{t^2}$$

$$t(v_t-v_0)=2[(s_t-s_0)-v_0t]$$

$$t(v_t+v_0)=2(s_t-s_0)$$

$$(s_t-s_0)=t\dfrac{v_t+v_0}{2}$$