Sign of the Acceleration

• The symbol for the acceleration " a " in physics can be positive, negative, or zero.

• In everyday language, we distinguish between a body having a positive acceleration and a negative acceleration by saying that the object is accelerating or decelerating. In physics, we say that a body has an acceleration that could be positive, negative, or even zero.

• It is not always correct to infer that if a body is decelerating that its speed is decreasing, unless you also mean that the positive direction of the frame of reference (which you are subconsciously using to describe the motion) is also in the same direction as the body is moving at that moment.

Free-Fall Example:
When you throw an object vertically upwards, its speed decreases as it goes up and increases as it comes down. Thus one might say that the object is decelerating on the way up and accelerating on the down. From this you might infer that it has no acceleration at all at the top of the motion, but that is not true. The first two statements are correct if you switch the frame of reference on the way down so that the positive direction is now down instead of up.

In physics we can use just one frame of reference to describe both the body's upward and downward motion. Choosing up as positive, we would say the object's acceleration is -9.80 m/s² both on the way up and the way down, and at the top of its motion.

On its way up its speed is increasing by 9.80 m/s every second in the downward direction. Since it is moving upwards initially, the numerical magnitude of its speed is getting smaller, and it is reasonable to say that the object is decelerating., or accelerating with a negative acceleration.

On the way down, the object's speed is still increasing by 9.80 m/s every second in the downward direction. Since it is moving downwards, the numerical value of its speed is getting bigger, thus leading to the statement that it is accelerating.

At the top of its motion, the object still has an non-zero acceleration. It does not momentarily go to zero. There are two ways that to show that the acceleration is not zero at the top of its motion.

1. At the top of its motion the speed of the object is zero. The acceleration is defined as the instantaneous change in an object's speed with time, a = dv/dt . If it has no speed and no change in speed, then it should just come to a halt and stay there when it reaches the top of its motion. It obviously does not say there, so it must change speed and have an acceleration. This argument is true no matter how small an time interval you pick near the top since the derivative is the limit as the time interval goes to zero.

2. A more physical argument come from the cause of the object's acceleration - the force of gravity. The force of gravity on the objects does not go to zero at the top of an object's motion, or at anywhere else. The force of gravity is always the same and is equal to the object's weight.