Key Principles of Free Fall and Real-World Problem Solving
Physics is one of the subjects which enjoys a diversity of opinions. There are students who just love physics and there are those who find it a little bit hard to understand. No matter which category you fall, the concept of Free Fall will surely stand before you in the exam.
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Definition of Free Fall
Free fall is the movement of an object or body only under the influence of gravity. The acceleration is caused by this external force on the object, hence the motion of the object will be accelerated. Thus, free-fall motion is also popularly known as acceleration due to gravity. The acceleration in this motion is constant because the gravitational force rather than the pull is downwards and has a constant value. And the scenario will even be the same when a body has zero gravity. For example, say that the body is thrown upwards. Hence, the term acceleration due to gravity means that the motion of an object under free fall with constant acceleration (g) towards the Earth can be calculated as,
g = 9.8m/s²
Motion Under Gravity Free Fall (Newtonian Mechanics)
The uniform gravitational field with zero air resistance: A small vertical distance close to the surface of the Earth where an object falls. As long as the air resistance is lower than the force of gravity on the body, or equivalently the terminal velocity of the body is much greater than the body’s velocity.
The initial velocity is v₀
The vertical velocity with respect to time is v(t)
The initial altitude is y₀
The altitude with respect to time is y(t)
The time elapsed is represented by t
The acceleration due to gravity (9.8 ms²near Earth’s surface) is denoted by g
v(t) = v0 - gt
y(t) = v0t + y0 - \[\frac{1}{2}\] gt2
The uniform gravitational field in the presence of air resistance: In the case of the uniform gravitational the mass of an object be considered as m, and the cross-sectional area be A, and if the Reynolds number is above the critical limit such that the v of the square of the fall velocity is proportional to the air resistance, has an equation of motion under gravity free fall
\[ m\frac{dv}{dt} = mg - \frac{1}{2} \rho\]CDAv2
[where the air density is represented by ρ, and the drag coefficient is represented with CD, which even though depends on the Reynolds number yet commonly popular as a constant].
Let us think of a body that was at rest but now falling with no change In the density of the air with altitude, then the equation will be
v(t) = \[v_{\infty} tanh \begin{pmatrix} \frac{gt}{v_{\infty}} \end{pmatrix}\],
Now when the terminal speed is provided, the equation will be,
\[v_{\infty} = \sqrt{\frac{2mg}{\rho C_{D}A}}\]
The vertical position of the body as a function of time can be found if the speed of the body against time can be integrated over time,
y = y0 - \[v_{\infty}^{2} ln cos h \begin{pmatrix} \frac{gt}{v_{\infty}} \end{pmatrix}\],
The uniform gravitational field with air resistance applies to parachutists and skydivers' motion while doing the activity and falling from a height.
Solved Examples
Question: Is the acceleration due to gravity or the value of ‘g’ a constant?
Let an object (say a ball) be dropped from a height on the surface of the Earth, but that height is minimal in comparison to the radius of the Earth (almost negligible).
The force acting during free fall motion is equal to the force of gravitation between the falling body and the Earth
Therefore we can put this in the form of an equation,
F = \[\frac{GMm}{(R + h)^{2}} \]
where R = radius of the Earth
where M = mass of the Earth ,and
G = the universal gravitation constant, and
we also assume that (R+h) is almost equal as R,
where R = radius of the Earth
∴F = \[\frac{GMm}{R^{2}}\] Equation(1)
And according to Newton’s second law, F = ma
And the acceleration due to gravity which is represented by g is also calculated by force per unit mass, then
F = mg Equation(2)
From equations (1) and (2) we can equate,
mg = \[G \frac{Mm}{R^2} \] or
g = \[G \frac{M}{R^2} \] Equation(2)
Therefore, from equation number 3, we can see that the value of 'g' is not as constant as indifferent to G's value, which is a universal constant. The value of the due to gravity depends upon the mass and radius of the object, and from which we can conclude that it will not be the same everywhere. But the freefall acceleration remains constant during the free fall motion. Therefore the equation of motion can be easily used if only the acceleration value is replaced by 'g' in all those equations.
Fun Facts
The motion of an object under free-fall: Long ago, Galileo discovered that all objects would experience the same ‘g,’ i.e., the acceleration due to gravity when the air resistance was not present. Later in 1971, astronaut D. Scott tried to experiment with this theory with practical proof as he on the Moon’s surface released a feather and a hammer from the same vertical length or height. The result was that the feather and the hammer both reached the ground of the Moon at the same time. This happened because, on the Moon, the gravitational pull is only one-sixth compared to that of Earth’s.
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Gearing up for the Finals
Final examinations are the reelection of one’s preparation and organizational abilities. The student must keep some of the tips in mind before going for the exams. These tips will keep them ahead of others.
A student must have gone through the previous year question before going for the previous year question papers and practice them thoroughly. One should ensure that the syllabus is complete, however, it doesn't hurt if the students leave out the unimportant part from the syllabus. Have a time management plan to tackle the question paper and remember to sleep well before the exam.
FAQs on Free Fall: Understanding the Physics and Applications
1. What exactly is free fall in Physics?
In Physics, free fall is defined as the motion of an object where the only force acting upon it is gravity. In this ideal state, all other forces, especially air resistance, are considered negligible. Any object that is dropped, or thrown vertically upwards or downwards, is in a state of free fall as soon as it is released.
2. What are some real-world examples of an object in free fall?
While a perfect free fall only happens in a vacuum, several real-world situations closely approximate it. Examples include:
- An apple detaching from a tree and falling to the ground.
- A stone dropped from a height.
- A spacecraft orbiting the Earth is in a continuous state of free fall, where its horizontal velocity keeps it from hitting the surface.
- A diver jumping off a diving board, after they have left the board and before they hit the water.
3. How are the equations of motion modified for an object in free fall?
The standard equations of motion are adapted for free fall by replacing the acceleration 'a' with the acceleration due to gravity (g), which is approximately 9.8 m/s² near the Earth's surface. The equations become:
- v = u + gt (First Equation)
- h = ut + (1/2)gt² (Second Equation, where 'h' is vertical distance)
- v² = u² + 2gh (Third Equation)
The sign of 'g' is taken as positive for downward motion and negative for upward motion.
4. Why do a heavy stone and a light feather fall at the same rate in a vacuum?
This happens because the acceleration of a freely falling object is independent of its mass. According to Newton's second law, acceleration is force divided by mass (a = F/m). The force of gravity on an object is its mass times the gravitational acceleration (F = mg). Therefore, a = mg/m. The mass 'm' cancels out, leaving a = g. This means both the heavy stone and the light feather, free from air resistance in a vacuum, will accelerate downwards at the same constant rate, 'g'.
5. How does air resistance affect an object that is supposedly in free fall?
In the real world, air resistance is an upward force that opposes the downward motion of a falling object. Unlike the ideal model of free fall, this force causes the object's acceleration to decrease as its speed increases. Eventually, the object may reach a terminal velocity, a constant speed where the upward force of air resistance perfectly balances the downward force of gravity, and the net acceleration becomes zero.
6. Is an object in free fall even when it's moving upwards?
Yes. Free fall refers to any motion under the sole influence of gravity. When you throw a ball upwards, from the moment it leaves your hand, gravity is the only significant force acting on it. Even though its velocity is upwards, its acceleration is constantly downwards at 9.8 m/s². This applies to its entire flight: on the way up, at its highest point (where velocity is momentarily zero), and on its way down.
7. What is the main difference between 'g' (acceleration due to gravity) and 'G' (Universal Gravitational Constant)?
Although related, 'g' and 'G' are fundamentally different concepts:
- Nature: 'g' is the acceleration an object experiences due to a celestial body's gravitational pull and is a vector. 'G' is a universal scalar constant that determines the inherent strength of the gravitational force between any two masses.
- Value: The value of 'g' varies with location (e.g., it's different on Earth, the Moon, or Mars). On Earth, it's about 9.8 m/s². The value of 'G' is constant throughout the universe, approximately 6.674 × 10⁻¹¹ N·m²/kg².
8. Must an object's initial velocity be zero for it to be in free fall?
No, this is a common misconception. Free fall begins when gravity is the only force acting on an object, regardless of its initial velocity. An object dropped from rest has an initial velocity of zero. However, an object thrown downwards or upwards has a non-zero initial velocity and is also in a state of free fall the moment it is released.
