A Physics Problem in Calculus.

Question

In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)$ as if it positive or negative.

Now, various references stated that if we let the upward direction to be positive then $a$ is negative and if we let downward to be the positive direction, then $a$ is positive. The problem in their claim is that they did not explain well how they arrived with that conclusion.

My question now is that, why is the acceleration $a$ negative if we choose the upward direction to be positive. Note: I need a simple but comprehensive answer. Thanks in advance.

Answer 1

The gravity force is downward and so is the acceleration (by $F=ma$).

So if you choose a downward axis, the acceleration is positive.

And if you choose an upward axis, the acceleration is negative.

Answer 2

You can model this on a linear one dimensional space, the "heigth" $x$ of the particle. Different forces are acting on the ball, on the one hand the gravitation, which is directed to the "ground", on the other hand in the beginning a force is applied towards the "top" as the ball is thrown in that direction. You can derive the equations of motion through Newtons formular $F(t)=m\ddot{x}(t)$ and the superposition of forces, the resulting force is the sum of each force.

Answer 3

Acceleration is defined as the derivative of the velocity, i.e. $a(t)=\frac{v(t)}{dt}$. When the ball is going upward, the speed of the ball decreases and thus the acceleration becomes negative.

Answer 4

The acceleration due to gravity is always downward. The convention is so.

Case I:Downward is the positive direction.

Let's us examine your case where the ball is falling vertically from the sky. Normally, we take the direction of initial motion of the ball to be positive. This happens in physics. Since the ball is accelerating downward due to gravity and it occurs in the direction of the initial motion then the acceleration is positive since we have choose downward as the positive direction. If there is a air resistance force acts in the ball. We know that air resistance occurs in the other direction of the object. So the air resistance must be negative!

The differential equation for an object undergoes acceleration can be modeled as follows.(We assume that mass is a constant variable)

$$m\frac{dv}{dt}=F_R$$

Where $F_R$ is a resultant force.

$$m\frac{dv}{dt}=F+kv$$

In our case

$F$ will be the $mg$ $k$ is the proportionality of the force

$kv$ or sometimes we use $kv^2$ instead to model the resistance acting on the object. We know that the resistance acts on the object opposes the direction of the object. Then we attribute it as

$$m\frac{dv}{dt}=F-kv$$

Case II Upward as the positive direction!

Again consider the object is falling downwards.We model our differential equation as. The weight of the object is acting downwards

$$m\frac{dv}{dt}=-F+kv$$

What we say about negative and positive of acceleration. Since acceleration are vectors. Vectors make up of magnitude and direction!

Can you see why case II is negative compared to case I.

I always think of $$ma=Upward+downward$$

Answer 5

Since it's a free fall, the acceleration is : $$\vec{a}(t) = \vec{g}$$

Since it is rectilinear you get :

$$a(t) = \vec{a}(t).\vec{z} =\vec{g}.\vec{z}$$

So if $\vec{g}$ and $\vec{z}$ have the same sign, i.e. downward, you get $a(t) = g$.

And if $\vec{g}$ and $\vec{z}$ have opposite sign, i.e. if $\vec{z}$ is upward as $\vec{g}$ is always downward, you get $a(t) = - g$.

It is more natural to take $\vec{z}$ upward because you will have positive $z(t)$ when the ball is up.

Answer 6

I will try to answer my own question but correct me if I am wrong. This answer was due to @Yves's answer and with the help of Serway's book which states that the negative in $a$ simply means that the acceleration is on the negative direction.

Clearly, if we set the upward direction to be positive, the gravitational force that acts on the object is in the negative direction. With the use of the Newton's law that $F=ma$, we have $(-)F=ma$. Now, since mass is a scalar quantity, in order for $F$ to be on the negative direction, $a$ must be on the negative direction. That is $a$ must be negative. Note, the negative in $a$ means the acceleration is also downward. Thanks for all your answers and comments.