PhysicsLAB Resource Lesson
Non-constant Resistance Forces

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In general, whenever a non-constant resistance force acts on an object integration methods must be used to calculate
  • the amount of work done by the force on the object or
  • a value for the object's instantaneous velocity.
The most common of this type of force is drag, or air resistance.
Horizontal Motion: A Car

Consider a vehicle that is moving along the road and is encountering a resistance force that is directly proportional to the car's velocity; F = -kv. While the driver applies pressure to the accelerator, the car is in a state of dynamic equilibrium; that is, the forward force supplied by the engine equals the resistance force supplied by the drag created by air resistance on the vehicle. If he removes his foot from the accelerator the car will begin to slow down.
We will now calculate an expression for the car's instantaneous velocity at any time t if its initial speed was vo.
Beginning with Newton's 2nd Law we can derive an expression for the car's acceleration.
But we can also express the acceleration as the rate of change of the car's velocity.
Setting these two expressions for acceleration equal to each other allows us to write a differential equation relating velocity and time.
Our next step is to separate variables so we can integrate.
Integrating both sides gives us
where C1 and C2 are integration constants and C3 = C2 - C1, another constant.
Raising both sides of our equation as powers of e yields the equation for velocity as a function of time. 
Setting our initial boundary conditions at time, t = 0, will now allow us to finish our derivation.
Our final equation is
Vertical Motion, A Projectile
In the presence of air resistance, projectiles no longer experience a constant acceleration.
  • While rising, their negative acceleration has a magnitude that is greater than 9.8 m/sec2since both gravity and air resistance are acting to slow them down.
  • A projectile released vertically will have an instantaneous acceleration equal to -9.8 m/sec2 while it passes through the apex.
  • Once projectiles begin falling, their negative acceleration has a magnitude that is less than 9.8 m/sec2 since gravity and air resistance oppose each other.
Projectile are said to reach terminal velocity when the magnitude of the air resistance they encounter equals their weight, AR = mg.
Air resistance is proportional to an object's cross sectional-area and its speed. In our case, our projectile's cross-sectional area will remain constant, so our equation will only examine how air resistance affects the projectile's instantaneous speed.
Refer to the following information for the next three questions.

A projectile is released from rest a distance h above the ground. you may assume that the air resistance it encounters is directly proportional to its velocity, AR = -kv. 
 Calculate the magnitude of projectile's terminal velocity.

 Describe the projectile's graph of v vs t.

 What is its equation for v(t) as it falls?

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