We consider only a vertical projectile motion against a quadratic drag force D = −c ∗ v ∗ v,
where v is the velocity of the projectile, and v is its speed.
(A) Calculate terminal velocity.
(B) Obtain equation for a vertical position change.
To calculate the terminal velocity, we need to find the value of velocity at which the net force on the projectile becomes zero. In other words, when the drag force becomes equal in magnitude to the gravitational force acting on the projectile.
(A) Terminal velocity can be determined by equating the drag force (D) to the weight of the projectile (mg), where m is the mass of the projectile and g is the acceleration due to gravity.
-D = mg
Since the drag force is given by -c * v * v, we can substitute this into the equation:
-c * v * v = mg
Rearranging the equation, we get:
v * v = -mg / c
Taking the square root of both sides, we obtain:
v = √(-mg / c)
This is the equation for the terminal velocity.
(B) To obtain the equation for vertical position change, we need to consider the forces acting on the projectile. In this case, we have two forces: gravity (mg) acting downwards, and the drag force (D) acting upwards.
The net force (F_net) acting on the projectile can be calculated as:
F_net = mg - D
Substituting the expression for the drag force (-c * v * v), we get:
F_net = mg + c * v * v
Using Newton's second law (F_net = ma), where a is the acceleration, we can write:
ma = mg + c * v * v
Since the projectile is moving vertically, the acceleration (a) is simply the second derivative of the displacement (y) with respect to time (t), i.e., a = d²y/dt². Also, the velocity (v) is the derivative of the displacement with respect to time, i.e., v = dy/dt.
Plugging in these values, we get:
m * d²y/dt² = mg + c * (dy/dt) * (dy/dt)
Rearranging the equation, we have:
m * d²y/dt² - c * (dy/dt) * (dy/dt) = mg
This is the equation for the vertical position change.