Question

A small sphere of mass 3.00 g is released from rest in a large vessel filled with oil, where it experiences a resistive force proportional to its speed. The sphere reaches a terminal speed of 6.00 cm/s.

What is the time constant?

a. 6.12e-3

b. 5.12e-3

c. 7.12e-3

d. 8.12e-3

What is the time at which the sphere reaches 80.0% of its terminal speed?

a. 5.85 ms

b. 9.85 ms

c. 11.9 ms

d. 7.85 ms

Answer 1

Solution:

Given:

mass (m) = 3 g = 0.003 kg

terminal speed (V_t) = 6 cm/s = 0.06 m/s

Since: terminal velocity is: V_t = mg / b

and, coefficient ( b ) = mg / V_t

b = (0.003 kg)(9.80 m/s²) / (0.06 m/s)

b = 0.49 kg/s

Therefore, the time constant τ is :

τ = m / b = (0.003 kg) / (0.49 kg/s) = 6.12 x 10^-3 s Answer : (B) 6.12 x 10^-3 s

Now: To find the time t it takes the sphere to reaches 80.0% of its terminal speed:

(0.80) V_t = V_t (1 - e^-t/τ)

1 - e^-t/τ = 0.80

e^-t/τ   = 0.20

-t/τ = ln(0.20)

-t/τ = - 1.609

t = 1.609 x (6.12 x 10^-3 s)

t = 9.854 x 10^-3 s = 9.85 ms