We consider only a vertical projectile motion against a quadratic drag force D= −c ∗ v ∗ v1 , where v1 is the velocity of the projectile, and v is its speed.

Question

We consider only a vertical projectile motion against a quadratic drag force D= −c ∗ v ∗ v1 , where v1 is the velocity of the projectile, and v is its speed.

(A) Calculate terminal velocity.

(B) Obtain equation for a vertical position change.

Answer 1

To calculate the terminal velocity of a projectile in vertical motion against a quadratic drag force, we need to find the velocity at which the drag force is equal in magnitude but opposite in direction to the gravitational force acting on the projectile.

(A) Terminal velocity is reached when the net force on the projectile becomes zero. In vertical motion, the net force is the difference between the gravitational force acting downward and the drag force acting upward. The gravitational force is given by the equation F_gravity = m*g, where m is the mass of the projectile and g is the acceleration due to gravity (approximately 9.8 m/s^2).

The drag force equation is given as D = -c*v*v1, where c is a constant and v1 is the velocity of the projectile. Since we are considering only vertical motion, we can assume v1 to be the magnitude of velocity, given by the equation v1 = |v|.

Setting the net force equal to zero, we have:

0 = m*g - c*v*v1

Simplifying the equation, we get:

c*v*v1 = m*g

Now, since v = |v|, we can write:

c*v*|v| = m*g

Rearranging the equation, we have:

v^2 = (m*g)/(c*|v|)

The magnitude of velocity is always positive, so we can remove the absolute value sign:

v^2 = (m*g)/(c*v)

Finally, rearranging the equation to solve for v, we get:

v^3 = (m*g)/c

Taking the cubic root of both sides, we find:

v = [(m*g)/c]^(1/3)

Thus, the formula for terminal velocity is:

v_terminal = [(m*g)/c]^(1/3)

(B) To obtain the equation for the vertical position change, we can use the kinematic equation for motion with constant acceleration:

Δy = v_initial*t + (1/2)*a*t^2

Here, Δy is the vertical position change, v_initial is the initial velocity, t is the time, and a is the acceleration.

In the case of a projectile in vertical motion, the only force acting is the gravitational force, resulting in a constant acceleration of -g (negative sign because it acts in the opposite direction to the positive vertical axis).

Considering the initial velocity to be v0 and the initial position to be y0, we can rewrite the equation as:

Δy = v0*t - (1/2)*g*t^2 + y0

This equation gives the vertical position change as a function of time for a projectile in vertical motion against a quadratic drag force.