Is it possible to calculate specific velocities from acceleration?

Question

I'm trying to understand a sample problem from a current (published 2011) high school physics textbook, but I really think the question is flawed. The task is to convert an acceleration-time graph (AT-graph) into a velocity-time graph (VT-graph). The AC-graph is given as:

Then the book gives the following instructions for finding velocities from accelerations:

Here's my first problem. Where did they get the formula $v=(\Delta a)(\Delta t)$? Isn't $\Delta a$ a more advanced concept called "jerk"? (a concept that this textbook never introduces, by the way). Shouldn't it be $\Delta v = a\Delta t$? And if so, we're not finding $v$ (which, correct me if I'm wrong, is impossible from the data given), but merely $\Delta v$.

Then they show the following calculations:

This chart is confusing to me. Is the time column an instant of time or a time interval? (I think it's a time interval). It seems to me that they're calculating $\Delta v$, not $v$, in which case it's impossible to draw a VT graph. But that's what they do next anyway:

Any feedback is appreciated.

Answer 1

You are correct, the right equation is $\Delta v = a \Delta t$. What do you expect from a high school textbook.

They are implicitly assuming $v(0) = 0$, so that $v(t) = v(t) - v(0) = \Delta v$.

The 1st column is instants in time.

Answer 2

The notation in the textbook is very confusing, but if you ignore the notational problems the book is correct. The book is trying to express the concept of an integral without calculus. For example the velocity is: $$v(t_0) = v(0) + \int_0^{t_0} a(t) dt $$ and they are trying to compute the integral as the area of the rectangle in the figure (and implicitly assuming $v(0)=0$). The confusion for people that know calculus is that $\Delta a$ and $\Delta t$ are usually thought of as very small changes in $a$ and $t$ (so that $\frac{\Delta a}{\Delta t}$ is an approximation to the "jerk" you mentioned: $\frac{da}{dt}$). However in the book the meaning of $\Delta a$ and $\Delta t$ is that they are the (large) sides of the rectangle. Very confusing but it is not wrong or a typo, just a very poor choice of notation.