Instantaneous velocity

Question

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.

Answer 1

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

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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.

Answer 2

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.)

Answer 3

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.

Answer 4

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

Answer 5

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.