# Instantaneous velocity

So here’s a question I’ve been thinking of for a while. Suppose we say, “an object is having an instantaneous velocity along a particular direction ( say 10 m/s along the $$x$$-direction)” . Is it fair to conclude that it is traveling in a straight line along the $$x$$-axis? Well my opinion on this is, For instance, a projectile ( on earth ) , the instantaneous velocity ( which is constant through out the journey ) is always in the $$x$$-direction while the body is executing a parabola in the $$x$$-$$y$$ plane? Please acknowledge me if I’m wrong.

Suppose we say, “an object is having an instantaneous velocity along a particular direction ( say 10 m/s along the x-direction)” . Is it fair to conclude that it is traveling in a straight line along the x axis?

No, you don't have enough information to conclude that.

Here's a simple example in which that conclusion would be false. Let the position of the object be given by

$$\vec{r} = (10\cdot t)\,\hat{\mathbf{x}} + t^2\,\hat{\mathbf{y}}\quad(\mathrm{m})$$

The instantaneous velocity of this object is then

$$\vec{v} \equiv \dot{\vec{r}} = 10\,\hat{\mathbf{x}} + (2\cdot t)\,\hat{\mathbf{y}}\quad\left(\mathrm{\frac{m}{s}}\right)$$

When $$t=0$$, the instantaneous velocity is 10 m/s along the x direction but the object is clearly not traveling in a straight line along the x axis

After some discussion in the comments, I do want to make clear that the last sentence above is stating two independent facts. For clarity, I'll state them in reverse order:

(1) The object is clearly not traveling in a straight line along the x axis (see that the $$y$$ coordinate is quadratic in $$t$$)

(2) When $$t=0$$, the instantaneous velocity of the object is 10 m/s in the direction

Thus, simply because “an object is having an instantaneous velocity along a particular direction ( say 10 m/s along the x-direction)”, it isn't valid to conclude that the it is traveling in a straight line along the x axis.

• Given your functions of position and velocity, the object most certainly is traveling in a straight line along the $x$ axis at $t=0$, since $\vec{v}(0) = 10\,\hat{\mathbf{x}}\quad\left(\mathrm{\frac{m}{s}}\right)$. – Josh Oct 1 '18 at 20:44
• @Josh, I honestly don't know how to respond to that. – Hal Hollis Oct 1 '18 at 20:48
• Meaning what? at $t=0$, there is no y-component of velocity, right? So it has only an x-component of velocity, i.e., it is moving -- instantaneously -- parallel to the x-axis. For any $t>0$ it will, of course, not be moving parallel to the x-axis...unless I'm missing something, in which case I anticipate your polite correction. – Josh Oct 1 '18 at 21:01
• @Josh, motion in a straight line is, well, motion in a straight line. I honestly don't know how to interpret your statement that the object is travelling in a straight line just at the moment $t = 0$. – Hal Hollis Oct 1 '18 at 21:13
• There's no interpretation necessary. It is physically and mathematically straightforward. The problem is that you're assuming the statement "moving in a straight line" to apply to a finite time interval, but the value of the function for a specific value of $t$ only tells you what is happening instantaneously. If the question is "is it moving in a straight line over any finite interval" then the answer is "no," but what you said was that "when $t=0$...the object is not travelling in a straight line along the x-axis," and that statement is false. – Josh Oct 1 '18 at 21:22

It is only fair to say "travelling in a straight line" when the acceleration vector points in the same direction (or directly opposite) as velocity. If you accept any looser definition then "instantaneous velocity" would seem to always be in a straight line.

In all of the examples given above, there is an acceleration component normal to the direction of motion, therefore the path is curved.

To generalize your problem (i.e. determine straight line motion in random directions)...

1. Compute both the instantaneous velocity AND instantaneous acceleration vectors. You need both of these.

2. Compute the cross product of velocity and acceleration.

3. Compare to zero. If the cross product is zero, the instantaneous velocity is in a straight line. (There are other ways to do this of course, but I believe this is computationally easiest.)

In a parabola, the instantaneous velocity isn't constant or only acting in one direction while travelling through it's path. It is always tangent to the parabola.

An object which has an instantaneous velocity in one direction does not have to keep travelling in that direction as long as there are other forces acting on it (for example, at the top of a parabola, the instantaneous velocity is purely horizontal, but then as soon as you go away from the top, there is a downwards component and it's not longer just in the x direction.

Instantaneous velocity is just what it's name suggests; the velocity of something at the exact instant you are analyzing it. With only information about the instantaneous velocity, that's not enough to say the velocity at another point in time, as forces can change the velocity of the object.

• My question is, from the following sentence, can I conclude the object is traveling in the x-direction only ? “ An object moving with an instantaneous velocity of x m/s in the x direction?” – EpicGamer123 Oct 1 '18 at 18:12
• @EpicGamer123 Are there other forces acting on it? I say multiple times that alone isn't enough to say for sure it is only travelling in one direction. The top of the parabola was an example of instantaneous velocity in the horizontal direction which doesn't only travel in the x-direction. – JMac Oct 1 '18 at 18:17
• @EpicGamer123, No. You can not make that conclusion. The velocity of a moving object can change with time, and if the only thing you know about a particle is it's velocity at one particular instant, then you don't know whether or how it's velocity might be changing. If you don't know how its velocity might be changing, then you can not know what path it will follow. – user205719 Oct 1 '18 at 18:17
• @EpicGamer123, I can tie an object to a 1 meter long string, and swing it in a circle at a constant speed of 2 m/s. It's instantaneous velocity (which has a magnitude AND a direction) is always 2 m/s, but its direction is continuously changing. – David White Oct 1 '18 at 18:32

Well a projectile(obliqe projectile moves with parabolic path and its velocity is not always along x axis. This happen only when it reaches its maximum height .At any time in the projectile motion its Velocity is given by v⃗ =v⃗ₓ+v⃗y where v⃗ₓ and v⃗y are the velocities along x axis and y axis respectively and this v⃗y changes with time because of gravitational force but v⃗ₓ remain contant because of no force(in vaccum) acting along x axis

The equation $$\overrightarrow{v}=\frac{\Delta_\overrightarrow{x}}{t}$$, with $$\overrightarrow{v}$$ being the velocity vector, $$\Delta_\overrightarrow{x}$$ the change in the position vector, and $$t$$ being the time passed only applies to a constant velocity. The more general equation for velocity would be $$\overrightarrow{v}=\overrightarrow{x}'(t)$$ with $$\overrightarrow{x}'(t)$$ being the derivative of the position vector with respect to time. So the instantaneous velocity is the derivative of the position vector with respect to time at that instant.