A small sphere of mass
m = 0.520 kg
is dropped from rest into a viscous liquid in which the resistive force on the sphere can be expressed as
FD = −bv,
and reaches one-fourth of its terminal speed in
3.14 s.
(a) What is the terminal speed of the sphere?
m/s
(b) What is the distance traveled by the sphere in
3.14 s?
To find the terminal speed of the sphere, we can use the fact that it reaches one-fourth of its terminal speed in 3.14 seconds. Terminal speed is the maximum speed reached by an object as it falls through a medium, when the force of gravity is balanced by the resistive force of the medium.
Let's denote the terminal speed as Vt, the resistive force as FD, and the velocity of the sphere as v.
The equation of motion for the sphere can be written as:
m * dv/dt = mg - bv
Here, dv/dt represents the rate of change of velocity (acceleration), and mg is the force due to gravity.
At terminal speed, the sphere experiences zero acceleration, so we can set dv/dt = 0 in the equation above.
Therefore, at terminal speed:
0 = mg - b * Vt
Now, we know that the sphere reaches one-fourth of its terminal speed in 3.14 seconds. Let's denote this initial velocity as v0.
At t = 0, v = v0
At t = 3.14, v = 1/4 * Vt
Using this information, we can solve for Vt.
First, let's calculate the initial velocity (v0) using the distance traveled in 3.14 seconds.
(b) Distance traveled by the sphere in 3.14 seconds:
To find the distance traveled, we need to calculate the initial velocity and then use it in the equation of motion.
The formula to calculate the distance traveled is given by:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
In this case, we need to find the initial velocity. We know that the sphere reaches one-fourth of its terminal speed in 3.14 seconds, so we can use this information to find the initial velocity.
Now, we can proceed to find the distance traveled by the sphere in 3.14 seconds.
I'll start with part (b) first.
To find the distance traveled (d) by the sphere in 3.14 seconds:
1. Calculate the initial velocity (v0) using the given information that the sphere reaches one-fourth of its terminal speed in 3.14 seconds:
v0 = 1/4 * Vt
2. Use the equation of motion to calculate the distance traveled (d) by substituting the values of initial velocity (v0) and time (t):
d = v0 * t + (1/2) * acceleration * t^2
Now, let's solve part (a) to find the terminal speed (Vt).
To find the terminal speed (Vt) of the sphere:
1. Rearrange the equation for the resistive force (FD) to isolate the terminal speed (Vt):
0 = mg - b * Vt
2. Substitute the given mass (m) and solve for Vt:
Vt = mg / b
Now, you can plug in the values given in the problem to find the terminal speed and the distance traveled by the sphere in 3.14 seconds.
I would never be able to type the math here. So print this example out, and work thru it. It is welldone.
https://www.physics.purdue.edu/webapps/index.php/course_document/.../6046