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In a viscous medium, consider a ball travelling under the influence of gravity and viscous force.

Is it always necessary that during its time in air, the ball attains terminal velocity

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  • $\begingroup$ Air....? Is this the same as the viscous medium, or another part of the trajectory? $\endgroup$ Commented May 21 at 13:56
  • $\begingroup$ Yeah same… I m assuming a standard case of ball falling through air and viscous force is acting on ball 6pietar*v .. $\endgroup$
    – Kampann
    Commented May 21 at 14:06

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In a simplified model such at drag being proportional to the speed $v$ or the squared speed $v^2$ the speed never reaches terminal velocity in general. Rather, it approaches the terminal velocity as $t\to\infty$. So no, it is not necessary that the ball attains terminal velocity in those situations.

Of course, in reality simplified models break down, and at some point the speed will be close enough to terminal velocity to where one could not reliably measure any difference between the ball's actual speed and terminal velocity.

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  • $\begingroup$ Just like, strictly speaking, a capacitor never fully charges... $\endgroup$
    – Jon Custer
    Commented May 21 at 16:47
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Yes.

The force of gravity is constant as an object falls. A drag force gets bigger when the object falls faster.

A falling object typically starts at speed $0$. There is no drag force at this speed. Gravity makes the object fall, giving it speed.

Gravity will make it fall faster and faster until the speed get big enough that the drag force is just a big as gravity. At that point, the force speeding it up is just as big as the force slowing it down. It keeps the same speed from there on.

As BioPhysicist says, theoretically it takes an infinite time for a speed that is almost fast enough to truly reach the terminal speed. But in practice, you can't tell the difference from terminal velocity after a few seconds.

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No, the ball can never exactly attain its terminal velocity. As the speed of ball increases, the viscous force also increase. So, as the ball gains velocity, its net force will decrease because net force is: force due to gravity subtracted by force due to viscosity. When the ball is near its terminal velocity, the force tends to 0. Thus, we can say that as the ball reaches terminal velocity, its rate of increase in velocity also decreases. This directs to the fact, that the ball can never ever reach its terminal velocity, it can only get close to it. Theoretically, it will take infinite amount of time to reach terminal velocity. Practically, we can reach close to terminal velocity (95%-99%) fast, in seconds.

If you need the formula for how long it will take to reach a certain fraction of terminal velocity, @G.Smith has given a wonderful derivation here- Is there a way to calculate the time taken by a falling object to reach terminal velocity?

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