Constant acceleration kinematics

[PFuser1232]

In cases like projectile motion, although the acceleration vector is constant (ignoring air resistance), a single coordinate axis fails to describe the motion of the particle, since it occurs in a plane; therefore we need a minimum of two coordinate axes to fully describe the trajectory of that particle. I was wondering, is there a case of constant acceleration where two axes would be insufficient to describe the trajectory (forcing us to use a three dimensional coordinate system)?

 

[Suraj M]

Maybe you could consider the trajectory situation in the presence of wind ➡ in the direction parallel to the ground and perpendicular to the original path of the trajectory!

 

[PFuser1232]

Maybe you could consider the trajectory situation in the presence of wind ➡ in the direction parallel to the ground and perpendicular to the original path of the trajectory!

The particle would deviate from its original trajectory along a curve and the motion would no longer occur in a single plane?

 

[Suraj M]

Exactly, it would look like a normal trajectory when viewed from the zy plane and xy plane. hence you need a 3D representation!

 

[jtbell]

But would the acceleration still be constant (in both magnitude and direction) in such a situation?

 

[PeroK]

In cases like projectile motion, although the acceleration vector is constant (ignoring air resistance), a single coordinate axis fails to describe the motion of the particle, since it occurs in a plane; therefore we need a minimum of two coordinate axes to fully describe the trajectory of that particle. I was wondering, is there a case of constant acceleration where two axes would be insufficient to describe the trajectory (forcing us to use a three dimensional coordinate system)?

If the acceleration vector is constant (both magnitude and direction), then I'd take that to be my -y axis!

How would you choose your x-axis?

 

[PFuser1232]

But would the acceleration still be constant (in both magnitude and direction) in such a situation?

If at t=0 the particle experiences two constant forces then the acceleration vector would be constant and along the resultant of the two forces, right?

 

[jtbell]

two constant forces

The force from air drag depends on the relative velocity of the object and the air.

 

[PFuser1232]

The force from air drag depends on the relative velocity of the object and the air.

I suppose this wasn't a very good example then. Back to my original question: is there a situation where a plane would cease to fully describe a trajectory where the acceleration is constant?

 

[A.T.]

is there a situation where a plane would cease to fully describe a trajectory where the acceleration is constant?

I don't think so. The plane containing the initial velocity and the constant acceleration vectors can never be left by the object.

 

[jtbell]

Suppose an object has initial velocity ##\vec v_0## and constant acceleration ##\vec a##. During a time interval ##\Delta t_1## it changes velocity by ##\Delta \vec v_1 = \vec a \Delta t_1## and has a new velocity ##\vec v_1 = \vec v_0 + \Delta \vec v_1##. The three vectors in this equation lie in a plane.

The object continues during a second time interval ##\Delta t_2##, changes its velocity by ##\Delta \vec v_2 = \vec a \Delta t_2##, and has a new velocity ##\vec v_2 = \vec v_1 + \Delta \vec v_2##.

Given that ##\vec a## is constant (in both magnitude and direction), can ##\Delta \vec v_2## and ##\vec v_2## lie outside the plane defined in the first paragraph?

 

[PeroK]

Another way to look at it, is to take one axis parallel to the acceleration and the other parallel to the component of the velocity perpendicular to the acceleration. That reduces things to 2D motion.

Even with a central force, you can choose the coordinates so that motion is 2D: choose the y-axis along the initial radial vector; and choose the x-axis parallel to the component of the initial velocity perpendicular to that.

 

[PFuser1232]

Suppose an object has initial velocity ##\vec v_0## and constant acceleration ##\vec a##. During a time interval ##\Delta t_1## it changes velocity by ##\Delta \vec v_1 = \vec a \Delta t_1## and has a new velocity ##\vec v_1 = \vec v_0 + \Delta \vec v_1##. The three vectors in this equation lie in a plane.

The object continues during a second time interval ##\Delta t_2##, changes its velocity by ##\Delta \vec v_2 = \vec a \Delta t_2##, and has a new velocity ##\vec v_2 = \vec v_1 + \Delta \vec v_2##.

Given that ##\vec a## is constant (in both magnitude and direction), can ##\Delta \vec v_2## and ##\vec v_2## lie outside the plane defined in the first paragraph?

##\Delta \vec{v}_2## and ##\vec{a}## are collinear, so ##\Delta \vec{v}_2## and ##\Delta \vec{v}_1## lie in the same plane. Adding them together yields a vector which has to lie in the same plane they are in. Therefore, all vectors lie in the same plane. Right?

 

[Suraj M]

The force from air drag depends on the relative velocity of the object and the air.

I didn't say air drag, i said wind in the horizontal direction perpendicular to the path of a normal trajectory! wouldn't that be in 3D?(OP's original question)?

 

[PWiz]

I didn't say air drag, i said wind in the horizontal direction perpendicular to the path of a normal trajectory! wouldn't that be in 3D?(OP's original question)?

The point is that with any constant wind flow, regardless of direction, a resultant acceleration vector can be found out (resultant of the acceleration due to wind flow and due to gravity), and this would again result in motion that can be described in a single plane. There is no global coordinate system here, so you can just arbitrarily redefine your x and y axes to describe the motion in the xy plane.

 

[Suraj M]

I'm sorry Pwiz but I'm not able to imagine the 3D situation. This is my interpretation, its probably wrong, but what's wrong in it?Could you point it out?

 

[PWiz]

I'm sorry Pwiz but I'm not able to imagine the 3D situation.
View attachment 79282

That is because the situation cannot be said to be in 3D if you redefine your axes. Try imagining that the object makes a parabolic path between the x, y and z axes somewhere. Now what if you "rotated" your view so that the parabolic path faces you? Would the motion still be 3 dimensional?

 

[jbriggs444]

The point is that with any constant wind flow, regardless of direction, a resultant acceleration vector can be found out (resultant of the acceleration due to wind flow and due to gravity), and this would again result in motion that can be described in a single plane. There is no global coordinate system here, so you can just arbitrarily redefine your x and y axes to describe the motion in the xy plane.

An important point being that the wind flow must be constant relative to the projectile's current velocity so that the resultant of gravity plus wind resistance will be constant, correct? The OP required constant acceleration. That entails that the wind velocity in the ground frame would need to be increasing at a steady pace both horizontally and downward so as to maintain a constant relative speed and, accordingly, a constant force on the projectile.

If the wind speed were held constant in the ground frame, the 3D trajectory would end by asymptotically approaching a line in a fixed direction depending only on gravity and wind and would start on a line in the direction of its starting velocity. Those two lines need not be parallel. They could be skew, thus not fitting into a single plane.

 

[PWiz]

An important point being that the wind flow must be constant relative to the projectile's current velocity so that the resultant of gravity plus wind resistance will be constant, correct? The OP required constant acceleration. That entails that the wind velocity in the ground frame would need to be increasing at a steady pace both horizontally and downward so as to maintain a constant relative speed and, accordingly, a constant force on the projectile.

If the wind speed were held constant in the ground frame, the 3D trajectory would end by asymptotically approaching a line in a fixed direction depending only on gravity and wind and would start on a line in the direction of its starting velocity. Those two lines need not be parallel. They could be skew, thus not fitting into a single plane.

By a constant wind flow, I meant a wind which exerts a constant force on the object(doesn't change direction as well), since only then can the resultant acceleration vector have a constant direction and magnitude resulting in planar motion, as you have pointed out.