Physical Significance of Numbers

In summary, when a ball is allowed to fall freely with a constant acceleration of 10 m/s^2, its velocity after traveling a distance of 10 m can be either +10 m/s or -10 m/s. This depends on the reference point and direction chosen to calculate the velocity. In the situation where the ball has a downward velocity, the velocity will be -10 m/s. However, if we consider a stationary reference point directly below the falling ball or a vertical take off of a rocket with an acceleration of 10 m/s^2, the velocity can also be +10 m/s. This second solution is possible and has a physical reality, and it can be explained by tracing the trajectory of the ball before it had
  • #1
gamemania1986
Consider this case: A ball is allowed to fall freely with a constant acceleration of 10 m/s^2. What will its speed be after traveling a distance of 10 m?

We get that:
v[initial] = 0 m/s
d = -5 m (downward direction is taken as negative)
a = -10 m/s^2

To find v[final], we will use the formula d = (v[final]^2-v[initial]^2) / (2a)

Rearranging, we will get:

v[final]^2 = 2ad + v[initial]^2

By plugging the numbers, we will get:

v[final]^2 = 100 m^2/s^2

By mathematics, there are 2 solutions to this problem, that is v[initial][1] = +10 m/s and v[initial][2] = -10 m/s. I can only think of the situation where the ball has a downward velocity. Is the other answer (the one with +10 m/s velocity) possible/does it has a physical reality? If so, can anyone explain the situation? Thanks a lot!
 
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  • #2
I think to obtain your first formula, the identity
vf - vi = a*t
is used.
Next, you divide by a to get t, and plug that into d = a/2 * t^2.

But strictly speaking, d, v, and a are vectors.
So you are in fact not allowed to divide by a.

You could work around this by saying you just talk about one component (the z-component) of each vector. But this implies that (vf-vi) has the same sign as a (i.e., minus).

Of course, this is still true in the answer.
 
  • #3
Hi,
u consider a stationary ball 'b' directly below the falling ball and calculate the velocity of 'b' in + with respect to free falling ball.

Else, if free fall is not crucial, vertical take off of a rocket (containing the ball) with acceleration 10 m/s^2. v = u + at = final velocity of both rocket & ball.
 
Last edited:
  • #4
By mathematics, there are 2 solutions to this problem, that is v[initial][1] = +10 m/s and v[initial][2] = -10 m/s. I can only think of the situation where the ball has a downward velocity. Is the other answer (the one with +10 m/s velocity) possible/does it has a physical reality? If so, can anyone explain the situation? Thanks a lot!


The -10 m/s velocity occurs when you trace the evolution of the system going forward in time. Consider the trajectory of the ball before it had the 0 m/s velocity.

Hurkyl
 
  • #5
Aaaah... I got it! Thanks a lot Hurkyl!
 

FAQ: Physical Significance of Numbers

What is the physical significance of numbers?

The physical significance of numbers refers to the meaning and relevance of numerical values in the context of a specific scientific or mathematical concept. It involves understanding the relationship between numbers and the physical world and how they are used to describe and measure various phenomena.

How do scientists determine the physical significance of numbers?

Scientists determine the physical significance of numbers through experiments, observations, and mathematical models. They use data and measurements to identify patterns and relationships between numbers and physical quantities, and then use these findings to interpret the meaning and significance of the numbers.

Why is understanding the physical significance of numbers important in science?

Understanding the physical significance of numbers is important in science because it allows us to accurately describe and explain natural phenomena. It also helps us make predictions and develop theories about the physical world, and enables us to communicate and collaborate with other scientists.

What are some examples of the physical significance of numbers in different fields of science?

In physics, numbers are used to describe fundamental physical quantities such as mass, velocity, and energy. In chemistry, numbers represent the number of atoms or molecules in a substance, as well as their properties and reactions. In biology, numbers are used to measure and describe characteristics of living organisms, such as size, growth rate, and genetic information.

How can understanding the physical significance of numbers help us in our daily lives?

Understanding the physical significance of numbers can help us make informed decisions in our daily lives. For example, understanding the significance of nutrition labels in terms of grams and percentages can help us make healthier food choices. It also allows us to interpret data and statistics presented in the media, helping us make more informed opinions and decisions.

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