Why is displacement directly proportional to acceleration?

Answer 1

Acceleration is given by the equation v-u/t, whereby v is the final velocity of a body, u is the initial velocity and t is the time. So for there to be acceleration, there has to be a change in initial velocity (the starting speed of the body) and the final velocity of the body or if there is no change it would be 0/t and hence the acceleration will be 0. And we know that with a change in speed there has to be a change in displacement i.e. s=d/t. So to sum all this up with an increase in acceleration there is an increase in distance. But this is a very tricky question because you asked about displacement and since displacement is a vector quantity, it has both magnitude and direction. So if a body accelerates forward, then its acceleration is directly proportional to its displacement. However, if the body stops and starts accelerating backwards, it becomes a whole new different story. Given that displacement has direction, if the body moves backwards, then the displacement will be negative, but the acceleration positive. So displacement is directly proportional to acceleration ONLY when a body is moving forwards, but displacement is indirectly proportional to acceleration when a body is moving backwards from it's point of rest. But for distance, given that it is a scalar quantity, it only has magnitude so it is not affected by the direction of movement of the body. So distance is directly proportional to acceleration when a body is moving both forwards and backwards.

Answer 2

It isn't. It is proportional to acceleration. This follows from momentum conservation which is a deeper law than Newton's second law (which implies the same of course, but Newton's second law is strictly not true at high speeds).

To give an even deeper, and possibly incomprehensible but still true, answer: momentum conservation is a result of the requirement that the laws of nature are the same at every point in space.

Answer 3

Acceleration is directly proportional to displacement in simple harmonic motion.

There are perhaps two good explanations for this, one technical and one intuitive.

First let us define simple harmonic motion.

When a particle moves in a straight line so that the displacement of the particle with time is exactly given by a simple sine (or cosine) of time, then that it is simple harmonic motion.

For example: x=A sine (w t) .

Answer 1: (In two steps)

(a) If we know position as a function of time, we know velocity is the time rate of change of position.

v = w A cosine (w t)

(b) If we know velocity as a function of time, we know acceleration is the time rate of change of velocity.

a = -w2 A sine (w t)

* So, acceleration is proportional to displacement, and a(t)=-w2 x(t).

Answer 2: (In three steps)

(a) Simple harmonic motion occurs when a mass on an ideal spring oscillates.

(b) From Newton's laws, we know that acceleration is directly proportional to force.

a=F/m

(c) We know the force of an ideal spring is proportional to displacement (F=-kx).

* So, acceleration is proportional to displacement, and a(t)= -k/m x(t).
(This also tells is that w2 =k/m.)

As a result, "acceleration is directly proportional to displacement in simple harmonic motion."

Answer 4

No. Distance is, but displacement has no direct relation to velocity.