How Long to Reach Half Terminal Speed with Air Resistance?

In summary, an object of mass m falls from rest with air resistance force directly proportional to its speed, given by F=-bv where b is a proportionality constant. The negative sign indicates the force is in the opposite direction of the velocity. To find the time it takes for the object to reach half of its terminal speed, we can use the equations a=g-bv/m and a=dv/dt, which can be combined to form a differential equation involving v and t. Solving this equation will give us the desired time.
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Homework Statement


An object of mass m falls from rest subject to air resistance force directly proportional to the object's speed, F=-bv, where b is the proportionality constant in N/(m/s) and the negative sign indicates the direction of the force opposite to the direction of the velocity. How long does it take this object to reach half of its terminal speed?


Homework Equations



a=g-bv/m, a=dv/dt, etc...

The Attempt at a Solution


tried, all failed.
 
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  • #2
Using the two equations you posted, and eliminating a between them, should give you a differential equation involving v and t. What is it? Can you solve it?
 

FAQ: How Long to Reach Half Terminal Speed with Air Resistance?

1. What is terminal speed?

Terminal speed, also known as terminal velocity, is the maximum velocity that an object can reach when falling through a fluid, such as air or water. It is when the force of air resistance is equal to the force of gravity, resulting in a constant speed.

2. How is terminal speed calculated?

The formula for calculating terminal speed is: Vt = √(2mg/ρACd), where Vt is the terminal speed, m is the mass of the object, g is the acceleration due to gravity, ρ is the density of the fluid, A is the projected area of the object, and Cd is the drag coefficient.

3. What role does calculus play in determining terminal speed?

Calculus is used to derive the equation for terminal speed by analyzing the forces acting on an object in motion. It allows us to calculate the rate of change of velocity with respect to time, which is essential in determining terminal speed.

4. How does time affect terminal speed?

The time it takes for an object to reach terminal speed depends on factors such as the object's mass, shape, and the fluid it is falling through. In general, the larger and more streamlined the object is, the faster it will reach terminal speed. However, once terminal speed is reached, it will remain constant regardless of how much time has passed.

5. What are some real-world applications of terminal speed and calculus?

Terminal speed and calculus are used in fields such as physics, engineering, and aerodynamics to understand and predict the behavior of objects in motion. For example, they are used in designing parachutes and determining the maximum safe speed for skydivers. They also play a crucial role in understanding and analyzing the motion of objects in free fall, such as in amusement park rides.

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