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Resistive force proportional to object velocity
At low speeds, the resistive force acting on an object that is moving a viscous medium is effectively modeleld as being proportional to the object velocity. The mathematical representation of the resistive force can be expressed as
`R = -bv`
Where v is the velocity of the object and b is a positive constant that depends onthe properties of the medium and on the shape and dimensions of the object. The negative sign represents the fact that the resistance froce is opposite to the velocity.
Consider a sphere of mass m released frm rest in a liquid. Assuming that the only forces acting on the spheres are the resistive froce R and the weight mg, we can describe its motion using Newton's second law. though the buoyant force is also acting on the submerged object the force is constant and effect of this force be modeled by changing the apparent weight of the sphere by a constant froce, so we can ignore it here.
Thus `mg - bv = m (dv)/(dt) rArr (dv)/(dt) = g - (b)/(m) v`
Solving the equation
`v = (mg)/(b) (1- e^(-bt//m))`
where e=2.71 is the base of the natural logarithm
The acceleration becomes zero when the increasing resistive force eventually the weight. At this point, the object reaches its terminals speed `v_(1)` and then on it continues to move with zero acceleration
`mg - b_(T) =0`
`rArr m_(T) = (mg)/(b)`
Hence `v = v_(T) (1-e^((vt)/(m)))`
In an experimental set-up four objects I,II,III,IV were released in same liquid. Using the data collected for the subsequent motions value of constant b were calculated. Respective data are shown in table.
`{:("Object",I,II,II,IV),("Mass (in kg.)",1,2,3,4),(underset("in (N-s)/m")("Constant b"),3.7,1.4,1.4,2.8):}`
If buoyant force were also taken into account then value of terminal speed would have

Options

A. decreased
B. increased
C. remained same
D. increased or decreased depending upon density of liquid.

Explanation

To solve the problem of determining how the terminal speed of a sphere changes when the buoyant force is taken into account, we can follow these steps:

### Step 1: Understand the Forces Acting on the Sphere
When the sphere is submerged in a liquid, the forces acting on it include:
- The weight of the sphere, \( mg \), acting downward.
- The buoyant force, \( F_B \), acting upward.
- The resistive force, \( R = -bv \), also acting upward (opposite to the direction of motion).

### Step 2: Write the Net Force Equation
The net force acting on the sphere can be expressed as:
\[ F_{\text{net}} = mg - F_B - bv \]
Where:
- \( F_B \) is the buoyant force.

### Step 3: Set Up the Equation for Terminal Velocity
At terminal velocity, the acceleration of the sphere becomes zero, which means the net force is also zero. Therefore, we can set the net force equation to zero:
\[ mg - F_B - bv_T = 0 \]
Where \( v_T \) is the terminal velocity.

### Step 4: Solve for Terminal Velocity
Rearranging the equation gives:
\[ mg - F_B = bv_T \]
Now, we can solve for the terminal velocity \( v_T \):
\[ v_T = \frac{mg - F_B}{b} \]

### Step 5: Analyze the Effect of Buoyant Force
The buoyant force \( F_B \) is equal to the weight of the liquid displaced by the sphere, which depends on the volume of the sphere and the density of the liquid. Since \( F_B \) is subtracted from \( mg \), including the buoyant force reduces the effective weight of the sphere. Thus, the terminal velocity \( v_T \) will be lower than it would be without considering the buoyant force.

### Conclusion
When the buoyant force is taken into account, the terminal speed of the sphere decreases because the effective weight of the sphere is reduced due to the upward buoyant force.