Physical Significance of Numbers

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SUMMARY

The discussion centers on the physical significance of velocity in free fall, specifically analyzing a ball falling under a constant acceleration of 10 m/s². Using the formula v[final]² = 2ad + v[initial]², participants derive that the final velocity can be +10 m/s or -10 m/s. The conversation highlights the importance of vector components in physics, emphasizing that the negative solution corresponds to downward motion, while the positive solution represents a hypothetical scenario. The discussion concludes with a clarification of the physical reality of both solutions in the context of motion.

PREREQUISITES
  • Understanding of kinematic equations, specifically v = u + at
  • Knowledge of vector components in physics
  • Familiarity with the concept of acceleration due to gravity
  • Basic mathematical skills for solving quadratic equations
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  • Study the implications of vector direction in kinematics
  • Explore the concept of free fall and its applications in real-world scenarios
  • Learn about the role of initial conditions in motion analysis
  • Investigate the differences between scalar and vector quantities in physics
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Students of physics, educators, and anyone interested in understanding the principles of motion and acceleration in classical mechanics.

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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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.
 
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:
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
 
Aaaah... I got it! Thanks a lot Hurkyl!
 

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