# How are acceleration, time and velocity related?

An object will have traveled a specific distance in a given amount of time, which is determined by the acceleration or velocity. Time is the medium through which both acceleration and velocity can occur.

Let's begin by defining velocity, which is simply "how much has this object moved (and in what direction) in a given time interval." Since velocity is a vector, it can be negative. Essentially, velocity is the rate of change of an object's displacement over its change in time.

If acceleration is negative, it is referred to as deceleration and describes something slowing down. In other words, something cannot speed up or slow down unless there is a specific change in velocity. Acceleration is a little more complicated; it is defined as the rate of change of the object's velocity over its change in time. In other words, "how much has the object's velocity changed in a given time interval."

Let's compare the two units: acceleration has meters/secondsquared, which simply means that it's not always moving the same distance in a given time; if the acceleration is positive, it's actually moving more distance as time goes by. Velocity has the SI units of meters/second, which makes sense because it's moving a certain amount of distance per a certain amount of time.

Another relationship between the two is that acceleration is zero when velocity is constant. This is because acceleration is defined as the change in velocity, and since there is no change in velocity, acceleration is zero. All of these comparisons can be seen in the graph below (if calculus isn't your thing, just concentrate on the diagram).

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Acceleration, time, and velocity are related through the fundamental equations of motion in physics. Specifically, acceleration is the rate of change of velocity with respect to time. In mathematical terms, acceleration (a) is equal to the change in velocity (Δv) divided by the change in time (Δt). This relationship is often expressed as:

a = Δv / Δt

Additionally, velocity (v) can be expressed as the integral of acceleration with respect to time:

v = ∫a dt

Furthermore, if acceleration is constant over a certain period of time, the relationship between velocity, acceleration, and time can be described by the equation:

v = u + at

Where:

- v represents final velocity
- u represents initial velocity
- a represents acceleration
- t represents time

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A jeep chases a thief with a constant velocity #v#. When the jeep is at distance d from the thief, he starts to run with a constant acceleration #a#. Show that the police will be able to catch the thief when #v >= sqrt(2ad)#?
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- Why is acceleration due to gravity constant?

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