Show that the speed of a moving particle over a time interval is constant if and only if its velocity and acceleration vectors are perpendicular over the time interval.

Explanation:

Let's start with the definition of a constant velocity.

If the velocity magnitude, in three dimensions,  is a constant value (C) we have a constant velocity, which means.

Now, we know that the dot product between v(t) and v(t) is the |v|².

If we take the derivative whit respect to time in both sides of this equation we will have:

We apply the product rule on the left side and the right side will zero because the derivative of a constant is 0.

We know that dv(t)/dt = a(t) (using the acceleration definiton)

Therefore, we conclude:

If the dot product is 0, it means that v(t) and a(t) are orthogonal.

I hope it helps you!

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step-by-step explanation: