Are suvat equations valid in 2 dimensional motion for constant acceleration?

[vcsharp2003]

Homework Statement

I saw the following in my book and it confused me. This was under the section on drift velocity of electrons In a conductor.

$\vec {v_1}= \vec {u_1} + \vec {a} t$

In the above equation, $\vec {v_1}$ is velocity of electron just before it collides with a fixed ion, $\vec {u_1}$ is the velocity of electron just after the last collision with another ion and $\vec {a}$ is the constant acceleration of the electron in question when a uniform electric field is applied to the conductor.

Relevant Equations

Suvat equations of motion

To my understanding, suvat equations must apply when motion is one dimensional and also the acceleration is constant pointing in the direction of motion or against the direction of motion. So I'm not sure about this.

Perhaps, the vector form just means that we can select an axes system (i.e. x and y axes) and resolve each vector in the suvat equation along this axes system; now we would have constant acceleration along x and y axes and we could apply the suvat equation to the x axis components and separately to the y axis components.

 

[kuruman]

To my understanding, suvat equations must apply when motion is one dimensional and also the acceleration is constant pointing in the direction of motion or against the direction of motion. So I'm not sure about this.

First of all your understanding is incomplete. Think of projectile motion. The SUVAT equations apply in this 2d case where the acceleration is not either in or against the direction of motion.

The drift velocity of electrons in a conductor is an average velocity in the direction of the acceleration which is opposite to the electric field established inside the conductor. See here for more details of the model.

 

[vcsharp2003]

First of all your understanding is incomplete.

So, it's correct to write the suvat equation in 2 dimensional motion or even 3 dimensional motion, provided the velocity, acceleration and displacement terms in the suvat equation are written as vectors. Right?

Was my second paragraph in my original question at attempting to explain this correct? I think so.

 

[Steve4Physics]

Perhaps, the vector form just means that we can select an axes system (i.e. x and y axes) and resolve each vector in the suvat equation along this axes system; now we would have constant acceleration along x and y axes and we could apply the suvat equation to the x axis components and separately to the y axis components.

'suvat' equations apply in any arbitrary direction providing the acceleration is constant in that direction.

Another way to look at it is this: if $\vec a$ is constant, then so are its components $a_x, a_y$ and $a_z$ (using Cartesian coordinates). We can then independently apply 'suvat' equations in $x, y$ and $z$ directions:
$v_x = u_x + a_xt$
$v_y = u_y + a_yt$
$v_z = u_z + a_zt$

If an electric field acts in the (say) $x$-direction, then the electric field accelerates a charged particle in the $x$-direction only; $a_y=a_z=0$. For example, when an electron moves through a metal with a field in the $x$-direction,, the electron is alternately accelerated (speeding up) by the field and slowed down by each collision with the lattice. The time-average $x$-component of velocity is the 'drift velocity' as already noted by @kuruman.

Minor edit.

 

[kuruman]

So, it's correct to write the suvat equation in 2 dimensional motion or even 3 dimensional motion, provided the velocity, acceleration and displacement terms in the suvat equation are written as vectors. Right?

Right, but with the provision that the acceleration is a constant vector, i.e. fixed magnitude and direction. If the acceleration is not constant, the SUVAT equations are not applicable.

Was my second paragraph in my original question at attempting to explain this correct? I think so.

It is correct, but note that a good choice of axes is to have one of them, say the x-axis, in the direction of the electric field. Then the acceleration is in that direction only which simplifies the description.

 

[haruspex]

it's correct to write the suvat equation in 2 dimensional motion or even 3 dimensional motion, provided the velocity, acceleration and displacement terms in the suvat equation are written as vectors. Right?

As others have pointed out, the SUVAT equations are valid if the acceleration vector is constant. More generally, $\Delta\vec v=\int\vec a.dt$, $\Delta \vec x=\int\vec v.dt$.

When $\vec a$ is constant, $\Delta\vec v=\int\vec a.dt=\vec a\int.dt=\vec a\Delta t=\vec a(t-t_0)$, $\Delta \vec x=\int\vec v.dt=\int(\vec{v_0}+\Delta\vec v).dt=\vec v_0\Delta t+\int\vec a(t-t_0).dt$
Taking the lower bound for t as zero:
$\Delta \vec x=\vec v_0 t+\vec a\int t.dt=\vec v_0 t+\frac 12\vec at^2$.