To derive the resultant displacement of the bullet using the given net force equation, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. Here's the step-by-step derivation:
Start with Newton's second law equation:
F_net = m * a
Substitute the given net force equation:
-kmv = m * a
Rearrange the equation to isolate the acceleration, a:
a = -kv
Recall that acceleration is the derivative of velocity with respect to time:
a = d(v)/dt
Rewrite the equation using differentials:
-kv dt = dv
Integrate both sides of the equation with respect to t:
-k ∫ dt = ∫ dv/v
Solve the integrals:
-kt + C1 = ln|v|
Where C1 is the constant of integration.
Exponentiate both sides of the equation:
e^(-kt + C1) = |v|
Since v is positive, we can drop the absolute value sign.
e^(-kt) * e^(C1) = v
Let's denote e^(C1) as K (a new constant).
e^(-kt) * K = v
Rearrange the equation to isolate e^(-kt):
e^(-kt) = v/K
Integrate both sides of the equation with respect to t:
∫ e^(-kt) dt = ∫ (v/K) dt
Solve the integrals:
-1/k * e^(-kt) + C2 = (v/K) * t + C3
Where C2 and C3 are constants of integration.
Rearrange the equation to isolate e^(-kt):
-1/k * e^(-kt) = (v/K) * t + C3 - C2
Let's denote (C3 - C2) as C, a new constant:
-1/k * e^(-kt) = (v/K) * t + C
Multiply both sides of the equation by -k:
e^(-kt) = -(v/K) * k * t - Ck
Simplify:
e^(-kt) = -kv * t - Ck
Rearrange the equation to isolate t:
t = -(1/k) * ln|-(v/K) * k * t - Ck|
Simplify the absolute value:
t = -(1/k) * ln((v/K) * k * t + Ck)
Rearrange the equation to isolate t:
(v/K) * k * t + Ck = -(e^(-kt) - 1) / t
Multiply both sides by (v/K) * k:
(v/K) * k^2 * t^2 + Ck^2 = -(v/K) * k * (e^(-kt) - 1)
Simplify:
(v/K) * k^2 * t^2 + Ck^2 = (1/K) * v * (1 - e^(-kt))
Rearrange the equation to isolate t:
(v/K) * k^2 * t^2 = (1/K) * v * (1 - e^(-kt)) - Ck^2
Divide both sides by (v/K) * k^2:
t^2 = (1/v) * (1 - e