Acceleration

 

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In physics, acceleration is defined as the rate of change of velocity, or as the second derivative of position (with respect to time). It is then a vector quantity with dimension length/time². In SI units, acceleration is measured in meters/second² (m/s²). The term "acceleration" generally refers to the rate of change in instantaneous velocity. (velocity is speed and direction)

In common speech, the term acceleration is only used for an increase in speed; a decrease in speed is called deceleration. In physics, any increase or decrease in speed is referred to as acceleration and similarly, motion in a circle at constant speed is also an acceleration, since the direction component of the velocity is changing.

Galileo's experiment with dropping heavy and light objects from a tower showed that all falling objects have the same motion, and his inclined-plane experiments showed that the motion was described by v=at+v_o. The initial velocity vo depends on whether you drop the object from rest or throw it down, but even if you throw it down, you cannot change the slope, a, of the v-t graph.

Since these experiments show that all falling objects have linear v-t graphs with the same slope, the slope of such a graph is apparently an important and useful quantity. We use the word acceleration, and the symbol a, for the slope of such a graph. In symbols, a=Δ v/Δt. The acceleration can be interpreted as the amount of speed gained in every second, and it has units of velocity divided by time, i.e., “meters per second per second,” or m/s/s. Continuing to treat units as if they were algebra symbols, we simplify “m/s/s” to read .” Acceleration can be a useful quantity for describing other types of motion besides falling, and the word and the symbol “a” can be used in a more general context. We reserve the more specialized symbol “g” for the acceleration of falling objects, which on the surface of our planet equals 9.8 m/s². Often when doing approximate calculations or merely illustrative numerical examples it is good enough to use g=10 m/s², which is off by only 2%.

 

The acceleration of gravity is different in different locations.Everyone knows that gravity is weaker on the moon, but actually it is not even the same everywhere on Earth, as shown by the sampling of numerical data in the following table.

location	latitude	elevation (m)	value of g (m/s2)
north pole	90oN	0	9.8322
Reykjavik, Iceland	64oN	0	9.8225
Fullerton, California	34oN	0	9.7957
Guayaquil, Ecuador	2oS	0	9.7806
Mt. Cotopaxi, Ecuador	1oS	5896	9.7624
Mt. Everest	28oN	8848	9.7643

 

The main variables that relate to the value of g on Earth are latitude and elevation. Although you have not yet learned how g would be calculated based on any deeper theory of gravity, it is not too hard to guess why g depends on elevation. Gravity is an attraction between things that have mass, and the attraction gets weaker with increasing distance. As you ascend from the seaport of Guayaquil to the nearby top of Mt. Cotopaxi, you are distancing yourself from the mass of the planet. The dependence on latitude occurs because we are measuring the acceleration of gravity relative to the earth's surface, but the earth's rotation causes the earth's surface to fall out from under you.

A typical neutron star is not so different in size from a large asteroid, but is orders of magnitude more massive, so the mass of a body definitely correlates with the g it creates. On the other hand, a neutron star has about the same mass as our Sun, so why is its g billions of times greater? If you had the misfortune of being on the surface of a neutron star, you'd be within a few thousand miles of all its mass, whereas on the surface of the Sun, you'd still be millions of miles from most of its mass.

 

Questions -

What is wrong with the following definitions of g?

(1) “g is gravity.”

(2) “g is the speed of a falling object.”

(3) “g is how hard gravity pulls on things.”

 

• When advertisers specify how much acceleration a car is capable of, they do not give an acceleration as defined in physics. Instead, they usually specify how many seconds are required for the car to go from rest to 60 miles/hour. Suppose we use the notation “a” for the acceleration as defined in physics, and “” for the quantity used in advertisements for cars. In the US's non-metric system of units, what would be the units of a

and ? How would the use and interpretation of large and small, positive and negative values be different for a as opposed to acar ad?

 

• A spacecraft (in space) with its rockets off (e.g. in a continuous orbit, or going up for some minutes, and then down)
• The Moon orbiting around the Earth.
• An object dropped in a vacuum tube for a physics demonstration at NASA's Zero-G Research Facility

Examples of objects not in free fall:

• Standing on the ground: the gravitational acceleration is counteracted by the normal force from the ground.
• Flying horizontally in an airplane: the wings' lift is also providing an acceleration.
• Jumping from an airplane: there is a resistance force provided by the atmosphere.

 

On Earth

Near sea level, an object in free fall in a vacuum will accelerate at approximately 9.81 m/s², regardless of its mass. With air resistance acting upon an object that has been dropped, the object will eventually reach a terminal velocity (around 120 mph (200 km/h) for a human body). Terminal velocity depends on many factors including mass, drag coefficient, and relative surface area, and will only be achieved if the fall is from sufficient altitude.

 

Homework Problems: Click here to get Homework Problems on acceleration for your practice.

 

 

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