السلفى
16-11-2005, 01:41 AM
بسم الله الرحمن الرحيم
TERMINAL VELOCITY
1.INTRODUCTION
1.1 GENERAL
When an object which is falling under the influence of gravity or subject to some other constant driving force is subject to a resistance or drag force which increases with velocity, it will ultimately reach a maximum velocity where the drag force equals the driving force. This final, constant velocity of motion is called a "terminal velocity", a terminology made popular by skydivers. For objects moving through a fluid at low speeds so that turbulence is not a major factor, the terminal velocity is determined by viscous drag.
2.CALCULATIONS
An object which is falling through the atmospher is subjected to two external forces One force is the gravitational force, expressed as the weight of the object. The other force is the air resistance, or drag of the object. The motion of any object can be described by Newton's second law of motion, force F equals mass m times acceleration a:
F = m * a
which can be solved for the acceleration of the object in terms of the net external force and the mass of the object:
a = F / m
Weight and drag are forces which are vector quantities. The net external force F is then equal to the diffreence of the weight W and the drag D
F = W - D
The acceleration of a falling object then becomes:
a = (W - D) / m
The drag force depends on the square of the velocity. So as the body accelerates its velocity and the drag increase. It quickly reaches a point where the drag is exactly equal to the weight. When drag is equal to weight, there is no net external force on the object and the object falls at a constant velocity as described by Newton's first law of motion. The constant velocity is called the terminal velocity .
We can determine the value of the terminal velocity by doing a little algebra and using the drag equation Drag depends on a drag cofficient Cd the air denisty r the square of the velocityV and some reference area A of the object:
D = Cd * r * V ^2 * A / 2
At terminal velocity, D = W. Solving for the velocity, we obtain the equation
V = sqrt ( (2 * W) / (Cd * r * A) )
where sqrt denotes the square root function Typical values of the drag coefficient are given on a separate slide
The terminal velocity equation tells us that an object with a large cross-sectional area or a high drag coefficient falls slower than an object with a small area or low drag coefficient. A large flat plate falls slower than a small ball with the same weight. If we have two objects with the same area and drag coefficient, like two identically sized spheres, the lighter object falls slower. This seems to contradict the findings of Galileo that all free falling objects fall at the same rate with equal air resistance. But Galileo's principle only applies in a vacuum, where there is NO air resistance and drag is equal to zero.
3.terminal velocity examples
Falling object Mass Area Terminal velocity
Skydiver 75 kg 0.7 m2 60 m/s 134 mi/hr
Baseball (3.66cm radius) 145 gm 42 cm2 33 m/s 74 mi/hr
Golf ball (2.1 cm radius) 46 gm 14 cm2 32 m/s 72 mi/hr
Hail stone (0.5 cm radius) .48 gm .79 cm2 14 m/s 31 mi/hr
Raindrop (0.2 cm radius) .034 gm .13 cm2 9 m/s 20 mi/hr
4.Hailstone Terminal Velocity
Contributing to the danger of large hailstones is the fact that they fall faster than small ones. That is, the terminal velocity increases with the size of the hailstone. Assuming the hailstones to be spherical and using a drag cofecient of C = 0.5 gives the following :
Radius (cm) v (km/hr) v(m/s) v(mi/hr)
.01 7 1.9 4.3
0.1 22 6.1 13.7
0.2 31 8.6 19.3
0.5 49 13.6 30.5
1.0 69.5 19.3 43.2
2.0 98.3 27.3 61
3.0 120 33.4 74.8
5.0 155 43.2 96.6
10.0 220 61 136
TERMINAL VELOCITY
1.INTRODUCTION
1.1 GENERAL
When an object which is falling under the influence of gravity or subject to some other constant driving force is subject to a resistance or drag force which increases with velocity, it will ultimately reach a maximum velocity where the drag force equals the driving force. This final, constant velocity of motion is called a "terminal velocity", a terminology made popular by skydivers. For objects moving through a fluid at low speeds so that turbulence is not a major factor, the terminal velocity is determined by viscous drag.
2.CALCULATIONS
An object which is falling through the atmospher is subjected to two external forces One force is the gravitational force, expressed as the weight of the object. The other force is the air resistance, or drag of the object. The motion of any object can be described by Newton's second law of motion, force F equals mass m times acceleration a:
F = m * a
which can be solved for the acceleration of the object in terms of the net external force and the mass of the object:
a = F / m
Weight and drag are forces which are vector quantities. The net external force F is then equal to the diffreence of the weight W and the drag D
F = W - D
The acceleration of a falling object then becomes:
a = (W - D) / m
The drag force depends on the square of the velocity. So as the body accelerates its velocity and the drag increase. It quickly reaches a point where the drag is exactly equal to the weight. When drag is equal to weight, there is no net external force on the object and the object falls at a constant velocity as described by Newton's first law of motion. The constant velocity is called the terminal velocity .
We can determine the value of the terminal velocity by doing a little algebra and using the drag equation Drag depends on a drag cofficient Cd the air denisty r the square of the velocityV and some reference area A of the object:
D = Cd * r * V ^2 * A / 2
At terminal velocity, D = W. Solving for the velocity, we obtain the equation
V = sqrt ( (2 * W) / (Cd * r * A) )
where sqrt denotes the square root function Typical values of the drag coefficient are given on a separate slide
The terminal velocity equation tells us that an object with a large cross-sectional area or a high drag coefficient falls slower than an object with a small area or low drag coefficient. A large flat plate falls slower than a small ball with the same weight. If we have two objects with the same area and drag coefficient, like two identically sized spheres, the lighter object falls slower. This seems to contradict the findings of Galileo that all free falling objects fall at the same rate with equal air resistance. But Galileo's principle only applies in a vacuum, where there is NO air resistance and drag is equal to zero.
3.terminal velocity examples
Falling object Mass Area Terminal velocity
Skydiver 75 kg 0.7 m2 60 m/s 134 mi/hr
Baseball (3.66cm radius) 145 gm 42 cm2 33 m/s 74 mi/hr
Golf ball (2.1 cm radius) 46 gm 14 cm2 32 m/s 72 mi/hr
Hail stone (0.5 cm radius) .48 gm .79 cm2 14 m/s 31 mi/hr
Raindrop (0.2 cm radius) .034 gm .13 cm2 9 m/s 20 mi/hr
4.Hailstone Terminal Velocity
Contributing to the danger of large hailstones is the fact that they fall faster than small ones. That is, the terminal velocity increases with the size of the hailstone. Assuming the hailstones to be spherical and using a drag cofecient of C = 0.5 gives the following :
Radius (cm) v (km/hr) v(m/s) v(mi/hr)
.01 7 1.9 4.3
0.1 22 6.1 13.7
0.2 31 8.6 19.3
0.5 49 13.6 30.5
1.0 69.5 19.3 43.2
2.0 98.3 27.3 61
3.0 120 33.4 74.8
5.0 155 43.2 96.6
10.0 220 61 136