# My physics teacher gave us this equation $v= -3 +3t$

She asked us if the body was accelerating or slowing down, and I immediately said that it was accelerating (because the $a=3>0$). Then she said that I was wrong because the direction of the acceleration vector was the opposite of the direction of initial speed($v_0=-3$). I do not understand why it slows down, because with the passage of time the body moves faster. Can someone give me an explanation?

• Clue: what is the velocity when $t = 1$? – John Rennie Feb 24 '15 at 15:11
• Technically speaking, you could go to your teacher and say that any change in velocity is classified as acceleration, so the question wouldn't make sense because it's doing both. Alternatively, you could say that in the body's initial inertial frame of reference it would be accelerating and the body only ever feels acceleration, and after $t=1$ everyone would say it's accelerating anyway, so the most generally correct statement would be that it is accelerating – Jim Feb 24 '15 at 15:47
• Of course, if you were my student and tried to pull that on me, I'd smack you for being a wise ass – Jim Feb 24 '15 at 15:47
• And if you were my student I would say you are exactly right (with the commend given by @JimdalftheGrey). That's not being a wise ass. That's using the term acceleration properly. If the teacher really wanted an unambiguous answer, she should have asked whether the body was getting faster or slower at some particular time. I'd vote down the teacher's question if I could. :) – Bill N Feb 24 '15 at 22:49

The initial velocity and acceleration here are in opposite directions. The magnitude of velocity (represented by $S=|\vec v|$) decreases upto a certain instant. (i.e. where $\vec v=0$).

Edit: Also, consider these graphs. ($t^.$ being the time where $v=0$) Note how the velocity increases but the magnitude of it (in the $S$, $t$ graph) decreases till $t^.$. • if i had to give only one answer, what should i choose: a) the body accelerates b)the body slows down ? – Al.Ka Feb 24 '15 at 15:27
• b) would be valid only in a given time period! The body does accelerate at all times. (slowing down is always related to a decrease in speed, not velocity.) – Hritik Narayan Feb 24 '15 at 15:29
• Our teacher told us that it slows down all the time. Thank you for the explanation. – Al.Ka Feb 24 '15 at 15:31

I understand what your teacher is saying, but I think she's wrong. In my physics classes we were told never to use the word "de"celeration, only acceleration. Why? For elegance reasons mostly.

• Acceleration is a vector quantity, therefore its magnitude is always positive or zero with a direction.

• The sign (or direction) of the acceleration depends on your reference frame. All inertial reference frames are equal and ultimately arbitrary; therefore to say a certain acceleration is DE-cceleration implies the reference frame you're working with is better, or somehow different, than the one where it would have been a mere ACCeleration. This is not elegant and we strive to have definitions that have elegant results.

For the last point imagine that what you are observing a tennis ball during a match. Then there is another observer observing from the other side of the court (i.e. bleachers on either side of the moving particle). In their reference frame the equation of motion would be: $v(t)=3-3t$. Can the particle really be decelerating, and accelerating? What's so special about her reference frame?

If anything DEceleration is layman's term to mean an acceleration that brings you to rest wrt to the ground. Which is not the case here after t=1

She asked us if the body was accelerating or slowing down

Acceleration is defined as the time rate of change of velocity and, in this example, the acceleration is constant and positive.

So, the full answer is: the velocity of the body is always increasing while the speed is decreasing for $t<1$ and increasing for $t>1$.

In this plot, the velocity $v$ is blue and the speed $|v|$ is red

$$v = -3 + 3t$$

$$|v| = |-3 + 3t|$$ The sense of acceleration has nothing to do with the sense of velocity. Bodies always have negative acceleration due to gravity regardless if they are going up or down. What is important is the convention as to which direction a positive displacement occurs.

In your case all you know that the acceleration vector is in the same direction as a positive displacement ($a=+3$) and your answer is correct.