# How Do You Calculate Time Ratios in Motion with Drag Force?

• tanaygupta2000
It would be better to use different symbols for these quantities.In summary, the conversation discusses the equations and calculations involved in finding the distance covered in terms of time taken, using the given equation F = -kdx/dt. It also highlights the importance of using different symbols to avoid confusion when dealing with multiple quantities.
tanaygupta2000
Homework Statement
At t=0, a particle of mass m having velocity u starts moving through a liquid kept in a horizontal tube and experiences a drag force F = -kdx/dt. It covers a distance L before coming to rest. If the times taken to cover the distances L/2 and L/4 are t2 and t4 respectively, then the ratio t2/t4 (ignoring gravity) is ?
Relevant Equations
F = -kdx/dt
Since given F = -kdx/dt
so I equated mx'' = -kx'
which gave x(t) = A + B exp(-kt/m)
hence v(t) = (-kB/m) exp(-kt/m)
and using v(0) = u, v(t) = u exp(-kt/m)

then I integrated dx = v(t)dt, dx from 0 to L and v(t)dt from 0 to t to find the distance covered L in terms of time taken t.
From this I got L = mu/k (1 - exp(-kt/m) )

Then I substituted L/2 = mu/k (1 - exp(-kt2/m) )
and L/4 = mu/k (1 - exp(-kt4/m) )

Now after this, I'm not getting how to find the ratio t2/t4 from the last two expressions of L/2 and L/4.

If it ultimately travels a distance L, then at infinite time, x = L

tanaygupta2000 said:
dx from 0 to L and v(t)dt from 0 to t to find the distance covered L in terms of time taken t.
You are in danger of confusing yourself by using the same symbol, L, for both the given distance covered at t=∞ and the distance covered at some arbitrary time t.

## 1. What is drag force and how does it affect motion?

Drag force is a type of resistance that occurs when an object moves through a fluid, such as air or water. It is caused by the friction between the object and the fluid, and it acts in the opposite direction of the object's motion. This force can slow down the object and change its direction.

## 2. How is drag force calculated?

The drag force on an object can be calculated using the equation Fd = 0.5 * ρ * v^2 * Cd * A, where ρ is the density of the fluid, v is the velocity of the object, Cd is the drag coefficient, and A is the cross-sectional area of the object. This equation takes into account the speed, size, and shape of the object as well as the properties of the fluid it is moving through.

## 3. How does the shape of an object affect drag force?

The shape of an object can greatly affect the amount of drag force it experiences. Objects with a streamlined shape, such as airplanes or fish, are designed to minimize drag and move through fluids with less resistance. On the other hand, objects with a large surface area and irregular shape, such as a parachute, will experience more drag force as they move through a fluid.

## 4. What is terminal velocity and how does it relate to drag force?

Terminal velocity is the maximum velocity that an object can reach when falling through a fluid due to the balance between the force of gravity and the drag force. As the object's speed increases, so does the drag force until it eventually equals the force of gravity. At this point, the object will stop accelerating and reach a constant speed, known as terminal velocity.

## 5. How can drag force be reduced?

There are several ways to reduce drag force on an object. One way is to decrease its speed, as drag force increases with velocity. Another way is to change the object's shape to a more streamlined design, which can reduce the surface area and decrease the friction with the fluid. Additionally, changing the properties of the fluid, such as using a lubricant, can also decrease drag force.

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