The acceleration of a freely falling object that is dropped from rest remains constant throughout the entirety of the fall. This acceleration is caused by gravity and is often represented by the symbol g. On Earth, the value of g is approximately 9.8 m/s2. This means a freely falling object near the surface of the Earth accelerates downward at 9.8 meters per second squared.
Quick Summary
For a freely falling object dropped from rest near the surface of the Earth:
- The initial velocity is 0 m/s
- The acceleration due to gravity is 9.8 m/s2 downward
- The acceleration remains constant throughout the fall
- Therefore, the acceleration at the end of the fall is 9.8 m/s2 downward
Deriving the Acceleration Equation
The acceleration of a freely falling object can be derived from Newton’s second law of motion and the law of gravity. Newton’s second law states that:
F = ma
Where F is the net force, m is the mass of the object, and a is the acceleration.
The law of gravity states that any two objects exert a gravitational force on each other. The magnitude of this force is given by:
F = Gm1m2/r2
Where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
For an object falling near the surface of the Earth, m1 is the mass of the Earth, m2 is the mass of the falling object, and r is approximately the radius of the Earth. The net force on the object is the gravitational force, so setting these two equations equal to each other gives:
Gm1m2/r2 = ma
Solving this for a gives:
a = Gm1/r2
Plugging in values for the mass and radius of the Earth gives the acceleration due to gravity, g, as:
g = 9.8 m/s2
So the acceleration of a freely falling object near the Earth’s surface is 9.8 m/s2 downward.
Kinematic Equations
The acceleration can also be derived from the kinematic equations of motion. For vertical motion with constant acceleration a and initial velocity v0, the equations are:
v = v0 + at
d = v0t + 1/2at2
v2 = v02 + 2ad
Where v is the final velocity, v0 is the initial velocity, t is time, and d is displacement. For a freely falling object dropped from rest, the initial velocity v0 is 0. Plugging this into the first equation gives:
v = at
This shows that for any point during the fall, the velocity v at that instant is equal to the acceleration a multiplied by the time t. Taking the derivative of velocity with respect to time gives acceleration:
a = dv/dt
For a freely falling object, velocity increases linearly with time at a constant rate given by the acceleration due to gravity. Therefore, dv/dt is constant and equal to the gravitational acceleration g. So again, this shows that a freely falling object has a constant downward acceleration g throughout the entire fall.
Independence of Mass
An interesting fact about falling objects is that their acceleration due to gravity does not depend on their mass. All objects fall with the same acceleration on Earth, neglecting air resistance. This can be seen by substituting F = mg into Newton’s second law:
F = ma
mg = ma
The mass cancels out, giving just:
a = g
So the mass has no effect on the acceleration. This independence of mass for falling objects was demonstrated in a famous experiment by Galileo Galilei. He showed that objects of different masses fall at the same rate if air resistance is negligible.
Effect of Air Resistance
One important caveat is that air resistance can affect the acceleration of falling objects. Air resistance applies a force opposite the direction of motion, slowing the acceleration. Air resistance depends on factors like the object’s speed, surface area, and shape. For example, a feather and hammer do not fall at the same rate on Earth because the feather experiences much more air resistance.
For typical falling objects like bricks or stones, air resistance is usually negligible unless the object is falling from a great height. Air resistance increases with speed, so it has a more pronounced effect on fast-moving objects. But for short falls near the Earth’s surface, the acceleration can be taken as g = 9.8 m/s2 with negligible error.
Examples
Let’s look at some examples to demonstrate these concepts:
Example 1
A ball is dropped from a height of 5 m above the ground. What is its acceleration at the moment it strikes the ground?
Solution:
The acceleration due to gravity is 9.8 m/s2 downward. This remains constant throughout the ball’s fall. Therefore, when the ball hits the ground, its acceleration is 9.8 m/s2 downward.
Example 2
A feather is dropped from a height of 2 m. What is its acceleration right before it hits the ground?
Solution:
Even though the feather experiences air resistance, just before striking the ground its velocity will be high. Air resistance depends on velocity – the higher the velocity, the greater the air resistance force. Just before hitting the ground, the feather reaches its terminal velocity, at which the air resistance force equals the gravitational force. At terminal velocity, the net force is zero and acceleration is zero. Therefore, just before hitting the ground, the feather’s acceleration is 0 m/s2.
Example 3
From the top of a tower 100 m high, a ball is thrown straight downward with an initial speed of 5 m/s. What is its acceleration after 3 seconds?
Solution:
The initial downward velocity is 5 m/s. With a constant downward acceleration of 9.8 m/s2 due to gravity, after 3 seconds the acceleration remains 9.8 m/s2 downward.
Using the kinematic equations:
v = v0 + at
v = 5 + 9.8(3) = 29.4 m/s
So after 3 seconds the velocity is 29.4 m/s downward and the acceleration is still 9.8 m/s2 downward.
Visual Representation
The acceleration of a freely falling object can be visualized on a graph of velocity vs. time. Since acceleration is the slope of the velocity-time graph, a constant downward acceleration due to gravity appears as a straight line with a constant negative slope:
| Time (s) | Velocity (m/s) |
|---|---|
| 0 | 0 |
| 1 | -9.8 |
| 2 | -19.6 |
| 3 | -29.4 |
| 4 | -39.2 |
The slope of this line is -9.8 m/s2, representing the constant downward acceleration.
Conclusions
In conclusion:
- Freely falling objects dropped from rest near Earth’s surface have an acceleration of g = 9.8 m/s2 downward
- This acceleration remains constant throughout the entire fall
- The acceleration is independent of mass and depends only on the gravitational field strength
- Air resistance can decrease the acceleration, but for short falls it is usually negligible
- The acceleration can be derived from Newton’s laws or calculated from kinematic equations
- The constant acceleration appears as a straight line with negative slope on a velocity-time graph
So for a freely falling object dropped from rest, the acceleration at the end of the fall is the same as the acceleration throughout the fall, which is 9.8 m/s2 downward near the Earth’s surface.
Applications
Understanding the motion of freely falling objects is useful for many applications including:
Physics Experiments
Measuring the acceleration due to gravity by timing falling objects or using motion sensors. Comparing different masses to illustrate independence of mass.
Engineering Design
Calculating impact forces and stresses for objects like bridges or amusement park rides. Designing systems to safely catch or decelerate falling objects.
Parachutes
Designing parachute systems involves balancing drag and gravity forces to reach a safe landing velocity and deceleration.
Trajectories
Analyzing the motion of projectiles like balls, rockets or planets by decomposing into horizontal and vertical (freely falling) motions.
Understanding Gravity
Freely falling motion provides insights into gravitational forces and the equivalence of gravitational and inertial mass.
Space Travel
Microgravity environments in space stations or capsules are achieved by freefall trajectories. Floating astronauts are actually freely falling.
Skydiving
Skydivers experience free fall acceleration until air resistance increases at higher speeds. Terminal velocity limits max speed.
Common Questions
Does acceleration increase during a free fall?
No, for an object falling freely near the Earth’s surface, the acceleration stays constant at 9.8 m/s2 downwards throughout the entire fall.
Does acceleration depend on an object’s mass?
No, all objects fall with the same acceleration due to gravity, regardless of their mass. Neglecting air resistance, a feather and hammer fall at the same rate.
What factors affect free fall acceleration?
The main factors affecting the acceleration are the gravitational field strength and any forces such as air resistance or buoyancy. Near the Earth’s surface, the field strength gives g = 9.8 m/s2.
Why do astronauts appear weightless in space?
Astronauts in orbit are actually just freely falling towards the Earth. They are accelerating downwards at 9.8 m/s2 but also moving sideways very fast, so they continually fall around the planet.
How does air resistance affect free fall?
Air resistance applies a force opposite to the motion, slowing the acceleration. This effect increases with velocity. It becomes noticeable for very light objects or falls from high altitude.
What techniques are used to measure free fall acceleration?
Measuring the time for an object to fall a known distance gives acceleration from physics equations. Optical gates, motion sensors, or high-speed cameras provide precise timing.
Practice Problems
Here are some practice problems to test your understanding:
Problem 1
A ball is dropped from a roof that is 30 m above the ground. What is the ball’s acceleration when it hits the ground?
Problem 2
An object falls off a 200 m cliff. How long does it take to reach the ground, assuming constant acceleration?
Problem 3
A firecracker explodes, sending a piece of debris straight upwards at 20 m/s velocity. What is the object’s acceleration 2 seconds later, assuming no air resistance?
Problem 4
Suppose acceleration during free fall increased over time like a = bt, where b is a constant. How would this affect the velocity-time graph?
Problem 5
Derive an expression for acceleration due to gravity in terms of Newton’s gravitational constant G, the mass of the Earth m, and the Earth’s radius R.
Summary
In summary, the acceleration of an object in free fall is constant throughout the fall at 9.8 m/s2 downwards, regardless of mass. This uniform acceleration results from the Earth’s gravitational field. Understanding free fall motion provides insight into gravity and projectile motion and has many applications in physics, engineering, and space travel.