Question

In a television commercial, a small, spherical bead of mass 4.00 g is released from rest at t=0 in a bottle of liquid shampoo. The terminal speed is observed to be 2.00 cm/s. Find (a) the value of the constant b in the equation v=mgb(1−e−bt/m), and (b) the value of the resistive force when the bead reaches terminal speed.

a) b=0.461s−¹, Resistive force = 0.0207 N
b) b=0.523s−¹ , Resistive force = 0.0185 N
c) b=0.384s−¹ , Resistive force = 0.0256 N
d) b=0.597s−¹, Resistive force = 0.0169 N

Answer 1

To find the value of the constant b, one can use the known conditions at terminal velocity and plug the given values into the v = mg/b (1 - e^-bt/m) equation, and then calculate the resistive force using F_resistive = mg - bv.

Step-by-step explanation:

To solve for the constant b in the exponential decay function v = mg/b (1 - e-bt/m), we can use the condition at terminal velocity where velocity v is constant and equals to the terminal speed. At terminal speed, acceleration is zero and the force of gravity is balanced by the resistive force, which allows us to ignore the exponential term.

Plug in the values of m (4.00 g), g (9.81 m/s2), and terminal speed v (2.00 cm/s) and solve for b to find its value. Once b is obtained, the resistive force at terminal speed can be calculated using the equation Fresistive = mg - bv, where v is the terminal velocity.