Question

please explain the answer

1) Up until now we have always ignored air resistance. We should now add it. Let us just think of simple 1-dimensional problem, dropping a ball of mass m from a height H but 2 with air resistance. We can model the air resistance as a force proportional to the velocity, fair = bv. The coefficient bis a constant. (For this problem you can use calculus textbooks or wolfram alpha to do the calculus.) What are the units on b? Why does it make sense for the force to be dependent on the velocity? The constant b depends on other physical parameters, can you think of what these might be? I only gave the magnitude of the air resistance force. What is the direction? Choose a standard coordinate system with the 3 direction being vertical and positive up. Write out Newton's second law for this problem. Since this is a 1-dimensional problem you don't have to worry about vector notation. When we derived our kinematic equations we integrated Newton's second law, essen- tially solving for the velocity and position from a known function for the acceleration. The kinematic equations assumed what abut the acceleration? How is this problem different? When the air resistance force and gravitational force balance out the acceleration goes to zero. When the acceleration goes to zero there is no longer a change in motion, hence the velocity does not change (right?). When this happens we say the object has reached terminal velocity, vy. Find an expression for the terminal velocity given the parameters of the problem. Following the procedure similar to the constant acceleration case, find the velocity as a function of time by integrating F = ma. Plot this function. Integrate your velocity v = dy/dt to get y as a function of time. Plot this function.

Answer 1

Answer : --

(a) since f = bv

so --

• dimension of force = dimension of (b v)
• dimension of b = $\frac{dim[F]}{Dim[v]}$
• $\frac{[ML^{-1}T^{-2}]}{[LT^{-1}]}$
• 
• So the unit of b = kgm^-2 s^-1 = Ns/m

(b) yes this resisting force is velocity dependent because it is a type of friction and as we know that the frictional force is come in play when there is relative motion occour . so this force is velocity depend .

this force is also depend on the weight f the falling object as well as the surface area of the object , velocity of the air , concentration of the medium etc.

(c) the direction of this force is opposite to that of the direction of motion of the object .

(d) the equation of the motion of the object is

$mg(-\hat{j})+bv\hat{j}= ma(-\hat{j})$