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# Acceleration of Gravity and Newton's Second Law

## Acceleration of gravity and Newton's Second Law - SI and Imperial units.

Acceleration of Gravity is one of the most used physical constants - known from

### Newton's Second Law

"Change of motion is proportional to the force applied, and take place along the straight line the force acts."

Newton's second law for the gravity force - weight - can be expressed as

W = Fg

= m ag

= m g                                 (1)

where

W, Fg = weight, gravity force (N, lbf)

m = mass (kg, slugs)

ag = g = acceleration of gravity (9.81 m/s2, 32.17405 ft/s2)

The force caused by gravity - ag - is called weight.

Note!

• mass is a property - a quantity with magnitude
• force is a vector - a quantity with magnitude and direction

The acceleration of gravity can be observed by measuring the change of velocity related to change of time for a free falling object:

ag = dv / dt                             (2)

where

dv = change in velocity (m/s, ft/s)

dt = change in time (s)

An object dropped in free air accelerates to speed 9.81 m/s (32.174 ft/s) in one - 1 - second.

• a heavy and a light body near the earth will fall to the earth with the same acceleration (when neglecting the air resistance)

### Acceleration of Gravity in SI Units

1 ag = 1 g = 9.81 m/s2 = 35.30394 (km/h)/s

### Acceleration of Gravity in Imperial Units

1 ag = 1 g = 32.174 ft/s2 = 386.1 in/s2 = 22 mph/s

### Velocity and Distance Traveled by a Free Falling Object

The velocity for a free falling object after some time can be calculated as:

v = ag t                          (3)

where

v = velocity (m/s)

The distance traveled by a free falling object after some time can be expressed as:

s = 1/2 ag t2                         (4)

where

s = distance traveled by the object (m)

The velocity and distance traveled by a free falling object:

Time
(s)
VelocityDistance
m/skm/hft/smphmft
1 9.8 35.3 32.2 21.9 4.9 16.1
2 19.6 70.6 64.3 43.8 19.6 64.3
3 29.4 106 96.5 65.8 44.1 144.8
4 39.2 141 128.7 87.7 78.5 257.4
5 49.1 177 160.9 110 122.6 402.2
6 58.9 212 193.0 132 176.6 579.1
7 68.7 247 225.2 154 240.3 788.3
8 78.5 283 257.4 176 313.9 1,029.6
9 88.3 318 289.6 198 397.3 1,303.0
10 98.1 353 321.7 219 490.5 1,608.7

Note! Velocities and distances are achieved without aerodynamic resistance (vacuum conditions). The air resistance - or drag force - for objects at higher velocities can be significant - depending on shape and surface area.

#### Example - Free Falling Stone

A stone is dropped from 1470 ft (448 m) - approximately the height of Empire State Building. The time it takes to reach the ground (without air resistance) can be calculated by rearranging (4):

t = (2 s / ag)1/2

= (2 (1470 ft) / (32.174 ft/s2 ))1/2

= 9.6 s

The velocity of the stone when it hits the ground can be calculated with (3):

v = (32.174 ft/s2) (9.6 s)

= 308 ft/s

= 210 mph

= 94 m/s

= 338 km/h

#### Example - A Ball Thrown Straight Up

A ball is thrown straight up with an initial velocity of 25 m/s. The time before the ball stops and start falling down can be calculated by modifying (3) to

t = v / ag

= (25 m/s) / (9.81 m/s2)

= 2.55 s

The distance traveled by the ball before it turns and start falling down can be calculated by using (4) as

s = 1/2  (9.81 m/s2) (2.55 s)2

= 31.8 m

### Newton's First Law

"Every body continues in a state of rest or in a uniform motion in a straight line, until it is compelled by a force to change its state of rest or motion."

### Newton's Third Law

"To every action there is always an equal reaction - if a force acts to change the state of motion of a body, the body offers a resistance equal and directly opposite to the force."

### Common Expressions

• mass loads: kg/m2 or kg/m3
• stress: N/mm2
• bending moment: kNm
• shear: kN
• 1 N/mm = 1 kN/m
• 1 N/mm2 = 103 kN/m2
• 1 kNm = 106 Nmm

### Latitude and Acceleration of Gravity

Acceleration of gravity varies with latitude - examples:

LocationLatitudeAcceleration og Gravity
(m/s2)
North Pole 90° 0' 9.8321
Anchorage 61° 10' 9.8218
Greenwich 51° 29' 9.8119
Paris 48° 50' 9.8094
Washington 38° 53' 9.8011
Panama 8° 55' 9.7822
Equator 0° 0' 9.7799

## Related Topics

• Basics - The SI-system, unit converters, physical constants, drawing scales and more.
• Dynamics - Motion - velocity and acceleration, forces and torque.
• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

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