motion

Question 1

How do you describe the motion of a particle launched from horizontal ground with velocity V

Answer 1

r = Vt + 1/2(gT^2),

where in component form,
x = Vtcosα
z = Vtsin(a) - 1/2(gT^2)

if you then rearrange for t in terms of x you can obtain the trajectory equation

Question 2

If the ground is horizontal then the range of the projectile, i.e. the
distance before hitting the ground, is

Answer 2

(V^2/g)(sin2a)

Question 3

when is the maximum height of the arc reached

Answer 3

when z’ = 0

which occurs at t =
(V/g)sin(a)

and is given by h = Vtsin(a) - 1/2gT^2

Question 4

we know t as a function of x, so when the pro- jectile lands at x = R the time spent in flight is

Answer 4

T = R/(Vcos(a) = (2V/g)sin(a)

Question 5

What is the envelope equation

Answer 5

gX2 tan^2 α − 2V^2X tan α + (2V^2Z + gX^2) = 0.

This quadratic has 3 possibilities.

• 2 real roots - two possible values for this particle to pass (X,Z)
• 1 repeated root - one value of a to hit (X,Z)
• no real roots so (X,Z) can’t be reached

Question 6

A reasonable mathemat-
ical model for the aerodynamic drag force on a particle is

Answer 6

Fd = -CV * abs(V)

Newton’s second law then becomes
mr ̈ = −mg − C|v|v