So now we get to see the use of all this math that we've been doing! Take the definition of acceleration eq. 3.7 and turn it around. Notice that now a is a constant (in time).
So we want to solve this equation for velocity v. v will be some function of time (because the system is accelerating), and we want to know what it is. So how do we solve
Such an equation is a simple example of a differential equation, because it involves a derivative and also an equals sign. That doesn't help us much in solving it, but at least we get to impress our friends with all this fancy math terminology. But to actually solve it isn't beyond the reaches of the human intellect.
From either way of looking at it you get, a straight line
Now I wrote v(t) because I wanted you to notice the the velocity depends on time, and not say, the color of the socks you're wearing. You don't have to write it this way, you're perfectly all right if you write it as just v, but then you have to remember what you're trying to solve for: how the velocity varies with time. Now C is an arbitrary constant. It could be anything with the information provided. So this is a bit strange. We're saying we don't know exactly what the solution to the equation is! The answer has this unknown constant floating around.
How can we determine that constant? Well the answer is, with the information given, we can't. In order to determine the constant, we need some additional piece of information.
A common type of problem is when you're given the initial value of a quantity at some time, say t = 0. This is called being given an ``initial condition'' . For example, you may say that initially the velocity of the object was 1.2 m/s. We'll see now that with an initial condition, the constant can be determined and we have a unique solution to this equation, in other words we have completely determined its velocity as a function of time.
The strategy here is to apply the initial condition to what we've just figured out, that is eq. 3.10. At t=0 we're saying we know that , some given initial velocity. So applying this we've got . So now we know C. It's just ! Plugging this into eq. 3.10, we have
You should be able to see that this answer makes sense intuitively. At t=0, the velocity starts out at , as we wanted it to. Then as time goes on, it increases linearly at a rate a.
Applying an initial condition, like we just did, is an important part of classical mechanics, and more generally the study of differential equations. It might seem weird at first, but you soon get used to it. The point is that depending on the initial conditions of a system, you're going to get different subsequent motion. If you toss a ball up into the air, then you're giving it one initial condition. If you throw it towards the ground, you're giving it another. What you see the ball do is different in these two cases, because they have two different initial conditions.