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When we project an object vertically upwards its velocity at maximum height consider to be zero but still, it has acceleration due to gravity i.e $9.8\mathrm{m/s^2}$. But mathematically we know that acceleration is the first derivative of velocity with respect to time $(a=dv/dt)$. So mathematically if velocity is zero there can not be any acceleration but this case contradict this equation. We can not even say that its direction is changing as the direction has no meaning without magnitude. So how this situation can be explained mathematically?

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Acceleration is defined as $$\vec{a}=\frac{d\vec{v}}{dt}$$

At maximum height $\vec{v}$ is zero but it doesn't means that $\frac{d\vec{v}}{dt}=0$ because at just the next moment i.e. $dt$ the particle gains some velocity i.e. $dv$ in downward direction.

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So mathematically if velocity is zero there can not be any acceleration

That is the wrong conclusion. Acceleration is a "change in velocity" not velocity itself.

Let's look at a simple example. Let's say we fire a projectile upwards an initial velocity of $100m/s$ and acceleration due to gravity is $10m/s^2$. That means that the velocity will be decreased by $10m/s$ for every $1s$ interval. So the velocity is $90m/s$ after $1s$, $80m/s$ after $2s$ etc.

After $10s$ it's $0m/s$ but it keeps decreasing: it's $-10m/s$ after $11s$. So while the velocity itself is $0$ at $10s$, the RATE of velocity change stays constant at $-10m/s$ per second.

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This can be a difficult concept at first. But notice the following facts:

  1. At the top of the trajectory, for an instant, the velocity is zero.
  2. An instant later, the velocity is non-zero. The particle is falling downward.

That means, the velocity has changed from zero to non-zero. What is acceleration but a change in velocity? If there were no change in velocity (no acceleration), the velocity would remain zero --- the particle would just kinda float stationary in the air. That's clearly nonsense.

Because the velocity doesn't stay at zero, then there must have been an acceleration.

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Suppose you walk past the point $x = 0$. At the moment you pass this point, your position is zero, but your velocity is nonzero. But your own logic would say this is impossible:

But mathematically we know that velocity is the first derivative of position with respect to time $(v=dx/dt)$. So mathematically if position is zero there can not be any velocity.

If you understand why it's possible to have zero position at some time, but nonzero velocity, then you'll understand why it's possible to have zero velocity at some time, but nonzero acceleration.

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