If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Worked example: motion problems (with definite integrals)

What can you say about the velocity and position of a particle given its acceleration. Created by Sal Khan.

## Want to join the conversation?

• What if the acceleration increases? Is there also a name for the derivative of acceleration in respect to time? •  There is actually a funny story behind this that involves cereal.

I believe the colloquialism began in Great Britain. As I understand it, the derivative of acceleration is jerk, the derivative of jerk is snap, the derivative of snap is crackle, the derivative of crackle is pop. Jerk and Jounce I think are the technical names of the fourth and fifth derivatives of position. However, I've heard tell that within the academic community there is some laughter over the fact that those derivatives (while rarely used outside of theoretical physics) are named after the Rice Crispies elves :P.
http://en.wikipedia.org/wiki/Snap,_Crackle_and_Pop
see the section of physics. :D
• Now I wonder: What quantity will we get by taking the derivative of acceleration? `a'(t) = ?` •  The derivative of acceleration is usually (and I am not making this up) called "jerk". It is called that because, if I understand correctly, a lack of uniform acceleration gives a vehicle a ride that "jerks". The term is also sometimes called the "surge".

The derivative of the jerk is called the jounce or the snap.
• What if I take the antiderivative of s(t)? Is there anything such as that? • Is there such a thing called a triple derivative? How would you apply it? • what is the integration of a constant?
(1 vote) • what does "c" represent? • The C represents an unknown constant value.

Consider:
Differentiation and integration are inverse processes, kind of like square root operator and the square operator, each "undoes" what the other did. But remember that if we have sqrt(x²) that there could be two solution since we don't know if the original x was negative or not since the square operation turns a negative into a positive, so in essence, we lost information when x was squared; we lost if it was originally positive or negative valued number.

Well, it is kind of the same with differentiation and integration.

Differentiate the following:
f(x) = x²,
f(x) = x² + 5,
f(x) = x² - 1000,
f(x) = x² + 185673
The derivative of all of them is f'(x) = 2x, right? We lost the constant value - we lost information about the original function f(x) when we took the derivative.

So now, when you take the indefinite integral of f'(x) = 2x, you do get back the x², but nothing comes back about any constant value that the function may also have had originally, so we say x² + C to account for the missing constant. The constant C might be zero, or it might be 5, it could be any value that was lost during the process of differentiation.
• I cannot understand this for some reason. I understand differentiation and integration but this concept baffles me. Watched this video multiple times but no clue, anyone got any tips? • shouldn't the acceleration be negative since velocity is get more and more negative . • It certainly should not. In order to explain why, we need to be clear about why the velocity is a negative number and what that means.
Velocity is a vector, which means that it has a magnitude (called speed) as well as a direction. In this particular case, there are only two relevant directions because we are working in a single-dimensional space. This basically means that we are only focussing on two directions like: Up and down or Right and left. The way that we differentiate between these two directions in a single-dimensional space is by the use of + and - signs. Plus usually represents up or right, while minus usually represents down or left. So the reason why the velocity is a negative number, is not because the object in motion is slowing down or anything like that - it simply means that the object is moving left or down. In addition, the absolute value of the velocity is going to be the speed (speed cannot be negative - you cannot drive negative 70 mph). So looking only at the magnitude of the number which represent velocity, you know that the object is speeding up if that absolute value increases as the variable t (time) increases.
In the video, Sal is showing an example of an object moving with constant accelleration. This means that we are looking at an object which keeps speeding up at a constant rate. This is why the velocity gets more and more negative - because the object is speeding up in a left or down direction.
If the acceleration had been (for example) -1, then the object would have deaccelerated at a constant rate and eventuelle come to a standstill and accellerate in the opposite direction (deacceleration is basically just acceleration in the opposite direction of motion). For example: If you throw an object into the air, then that object is constantly deaccelerated (or accelerated in the opposite direction; towards earth) by the force of gravity. Due to this force, the object will eventuelly cease its upwards motion and come to a halt in midair before dropping back down to earth (accelerated towards earth).  