Question

What are the 5 SUVAT equations?

Answer 1

The 5 SUVAT equations are a set of kinematic equations used to solve problems involving uniformly accelerated motion, linking displacement, initial and final velocities, acceleration, and time.

Step-by-step explanation:

The term 'SUVAT' is an acronym representing five variables in kinematics: displacement (S), initial velocity (U), final velocity (V), acceleration (A), and time (T). These variables are used in a set of equations to solve problems involving uniformly accelerated motion. The five SUVAT equations are as follows:

1. v = u + at - This equation links the final velocity (v), initial velocity (u), acceleration (a), and the time (t) taken.
2. s = ut + (1/2)at2 - This determines the displacement (s) using the initial velocity, acceleration, and time.
3. s = vt - (1/2)at2 - Similar to the second equation but uses the final velocity instead of the initial.
4. s = ((u + v)/2)t - This calculates displacement by averaging initial and final velocities and multiplying by the time.
5. v2 = u2 + 2as - Connects the squared final velocity to the squared initial velocity, acceleration, and displacement.

These equations form the cornerstone for solving motion problems where the acceleration is constant, enabling physicists to predict future motion given a set of initial conditions.

Answer 2

Here are the 5 SUVAT equations:

`\boxed{\begin{array}{ccc} \text{\underline{The Five SUVAT Equations:}} \\\\ 1. \ s = ut + (1)/(2)at^2 \\\\ 2. \ v = u + at \\\\ 3. \ v^2 = u^2 + 2as \\\\ 4. \ s = (1)/(2)(u + v)t \\\\ 5. \ s = vt - (1)/(2)at^2 \end{array}}`

Step-by-step explanation:

The SUVAT equations are a set of five equations used in classical mechanics to describe the motion of objects under constant acceleration. They are derived using the principles of kinematics and are applicable when acceleration is constant.

The term "SUVAT" comes from the variables used in the equations:

• 'S' for displacement
• 'U' for initial velocity
• 'V' for final velocity
• 'A' for acceleration
• 'T' for time

Here's a derivation for each of the five equations:
`\hrulefill`

Equation (1)
`\hrulefill`

We know acceleration is the rate of change of velocity, 'v', so we can say:

`\Longrightarrow a=(dv)/(dt)`

Multiplying both sides by 'dt', we get:

`\Longrightarrow adt=dv`

Now we can integrate both sides with the following limits:

`\displaystyle \Longrightarrow \int\limits^t_0 {a} \, dt =\int\limits^v_u {} \, dv \ \Big(\text{Remember, '$a$' is held constant}\Big)\\\\\\\\\Longrightarrow \Big[at\Big]\limits^t_0 = \Big[v\Big]\limits^v_u\\\\\\\\\Longrightarrow \Big[at-0\Big] = \Big[v-u\Big]\\\\\\\\\therefore \boxed{v=u+at}`

Thus, equation (1) is found.

`\hrulefill`

Equation (2)
`\hrulefill`

We know velocity is the rate of change of displacement, 's', so we can say:

`\Longrightarrow v=(ds)/(dt)`

Multiplying both sides by 'dt', we get:

`\Longrightarrow vdt=ds`

Now we can integrate both sides with the following limits:

`\displaystyle \Longrightarrow \int\limits^t_0 {v} \, dt =\int\limits^(s_f)_(s_0) {} \, ds`

Plugging in equation (1) for 'v':

`\displaystyle \Longrightarrow \int\limits^t_0 {(u+at)} \, dt =\int\limits^(s_f)_(s_0) {} \, ds\\ \\\\\\\Longrightarrow \Big[ut+(1)/(2)at^2\Big]\limits^t_0 = \Big[s\Big]\limits^(s_f)_(s_0) \\\\\\\\\Longrightarrow \Big[ut+(1)/(2)at^2-0\Big] = \Big[s_f-s_0\Big] \\\\\\\\\therefore \boxed{s=ut+(1)/(2)at^2}`

Thus, equation (2) is found.

`\hrulefill`

Equation (3)
`\hrulefill`

Start by taking equation (1):

`\Longrightarrow v=u+at`

Square both sides of the equation:

`\Longrightarrow v^2=(u+at)^2\\\\\\\\\Longrightarrow v^2=(u+at)(u+at)\\\\\\\\\Longrightarrow v^2=u^2+2uat+a^2t^2`

Notice that equation (2) shows up on the right-hand side if we factor out '2a':

`\Longrightarrow v^2=u^2+2a\left(ut+(1)/(2)at^2\right)`

`\therefore \boxed{v^2=u^2+2as}`

Thus, equation (3) is found.

`\hrulefill`

Equation (4)
`\hrulefill`

Lets take equations (1) and (2):

`\left\{\begin{array}{ccc}v=u+at & \dots (1)\\s=ut+(1)/(2)at^2 & \dots (2)\end{array}`

Solve equation (1) for 'a':

`\Longrightarrow v=u+at\\\\\\\\\Longrightarrow at=v-u\\\\\\\\\therefore a =(v-u)/(t)`

Now substitute this equation into equation (2):

`\Longrightarrow s=ut+(1)/(2)at^2 \\ \\ \\ \\ \Longrightarrow s=ut+(1)/(2)\left((v-u)/(t)\right)t^2 \\\\\\\\\Longrightarrow s=ut+(1)/(2)\left(v-u\right)t\\\\\\\\\Longrightarrow s=ut+(1)/(2)vt-(1)/(2)ut\\\\\\\\\Longrightarrow s=(1)/(2)vt+(1)/(2)ut\\\\\\\\\therefore \boxed{s=(1)/(2)(v+u)t}`

Thus, equation (4) is found.

`\hrulefill`

Equation (5)
`\hrulefill`

Taking equation (1) and rearranging:

`\Longrightarrow v=u+at\\\\\\\\\therefore u=v-at`

Plug the above into equation (2):

`\Longrightarrow s=ut+(1)/(2)at^2\\\\\\\\\Longrightarrow s=(v-at)t+(1)/(2)at^2\\\\\\\\\Longrightarrow s=ut+(1)/(2)at^2\\\\\\\\\Longrightarrow s=vt-at^2+(1)/(2)at^2\\\\\\\\\therefore \boxed{s=vt-(1)/(2)at^2}`

Thus, equation (5) is found.

`\hrulefill`

Thus, all five SUVAT equations have been derived, here they are listed below:

`\boxed{\begin{array}{ccc} \text{\underline{The Five SUVAT Equations:}} \\\\ 1. \ s = ut + (1)/(2)at^2 \\\\ 2. \ v = u + at \\\\ 3. \ v^2 = u^2 + 2as \\\\ 4. \ s = (1)/(2)(u + v)t \\\\ 5. \ s = vt - (1)/(2)at^2 \end{array}}`