Calculating Particle Speed with Velocity-Dependent Force

In summary, the speed of a particle with mass m moving parallel to the x-axis and actuated by a speed-dependent force directed against its movement is given by v(t) = v0 * cos(bt + ln(m) + c), where b is a positive constant (N units), m is the mass, t is time, c is another positive constant, and v0 is the initial velocity of the particle. This expression was obtained by integrating the given force equation and using the identities for natural logarithm and exponential functions.
  • #1
ssmooc
3
0
A particle of mass m moving parallel to the x-axis is actuated by a speed-dependent force directed against its movement. The force is given by:

F → = -be ^ CV I

where b is a positive constant (N units), and IA also a positive constant (units of s-1 m) and v is the velocity, the magnitude of the particle velocity (unit m · s -1).

If t = 0, the particle moves with velocity v0. Find the speed v (t) as a function of time t. Express your answer in terms of some or all of the following: b, m, t, c, v0. Use ln () for the natural logarithm function and e ^ () to the exponential function, if necessary.

v (t) =?

2. Homework Equations

v (t) =

3. The Attempt at a Solution

v (t) = (v_0 * b * cos (* t v_0) v_0 * + C * cos (* t v_0))
 
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  • #2
ssmooc said:
A particle of mass m moving parallel to the x-axis is actuated by a speed-dependent force directed against its movement. The force is given by:

F → = -be ^ CV I

where b is a positive constant (N units), and IA also a positive constant (units of s-1 m) and v is the velocity, the magnitude of the particle velocity (unit m · s -1).

If t = 0, the particle moves with velocity v0. Find the speed v (t) as a function of time t. Express your answer in terms of some or all of the following: b, m, t, c, v0. Use ln () for the natural logarithm function and e ^ () to the exponential function, if necessary.

v (t) =?

2. Homework Equations

v (t) =

3. The Attempt at a Solution

v (t) = (v_0 * b * cos (* t v_0) v_0 * + C * cos (* t v_0))
please show your working.
 

Related to Calculating Particle Speed with Velocity-Dependent Force

1. What is velocity dependent force?

Velocity dependent force is a type of force in physics that changes in magnitude or direction based on the velocity of an object. This means that the amount of force experienced by an object is dependent on its speed.

2. How does velocity dependent force affect motion?

Velocity dependent force can cause changes in an object's motion. For example, if an object is experiencing a velocity dependent force in the opposite direction of its motion, it can slow down or come to a stop. On the other hand, if the force is in the same direction as the object's motion, it can cause the object to speed up.

3. What is an example of velocity dependent force?

One example of velocity dependent force is air resistance. As an object moves through the air, the force of air resistance increases with the object's velocity. This is why it is easier to bike or run at a slower speed compared to a faster speed, because the air resistance is lower at slower speeds.

4. How is velocity dependent force different from a constant force?

A constant force is one that does not change in magnitude or direction, regardless of an object's velocity. On the other hand, velocity dependent force changes based on the object's speed. This means that a constant force will have the same effect on an object's motion, while velocity dependent force can have different effects based on the object's velocity.

5. What are some applications of velocity dependent force in the real world?

Velocity dependent force plays a significant role in many real-world applications, such as aerodynamics, vehicle design, and sports. Understanding how velocity dependent force affects objects in motion is crucial for designing efficient and safe vehicles, predicting the trajectory of projectiles, and improving athletic performance.

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