For a projectile of mass mo fired with initial velocity vo, if there is no drag

Question

For a projectile of mass mo fired with initial velocity vo, if there is no drag force: a) What is the trajectory r(t) of the projectile in polar coordinates. (Find r(t) and (t)) b) Using a), find the firing angle 0i that maximizes the projectile's time of flight.(Must show work) c) Using a), find the firing angle , that maximizes the projectile's range. For a 2 cm diameter, 1 kg projectile launched through air (STP) at -/4 with velocity vo - 15m/s, if the drag force is linear: d Calculate the energy dissipated by air resistance during the time of flight using 0 e Calculate the change in the kinetic energy Ko2 between the initial launch and impact * Solve d) and e) using either Cartesian or polar coordinates. There is not an analytic solution for this part

Explanation / Answer

given projectile

mass mo, initial velocity vo

a) at time t

x = (vo.i) t

y = (vo.j)*t - 0.5gt^2 ( where i and j are unit vectors in cartesian coordinates)

y = (vo.j)x/(vo.i) - 0.5gx^2/(vo.i)^2

now

in polar coordinates

r(t) = sqrt(x^2 + y^2) = sqrt((vo.i)^2*t^2 + ((vo.j)*t - 0.5gt^2)^2)

theta(t) = arctan(y/x) = arctan(((vo.j)t - 0.5gt^2)/(vo.i)t)

b) for maximising time of flight

(vo.i)t*tan(theta) = (vo.j)t - 0.5gt^2

t! = 0

tan(theta) = 0 at t = T

T = 2(vo.j)/g

for maximising this

vo.j = |vo|

firing angle is thetat = 90 deg

c) for maximising range

dr/dt = 0

(vo.i)^2*2t + 2((vo.j)^2*t - 0.5gt^2)((vo.j)^2 - gt) = 0

((vo.i)^2*2 + 2(vo.j)^4) - t(2g(vo.j) + g(vo.j)^2) + g^2t^2 = 0

solving we get

T = 2(vo.j)(vo.i)/gvo

hencve

this is true for thetat = 45 deg

d) for d = 2 cm, m = 1 kg

thetao = pi/4

vo = 15m/s

Fd = -kv

so,

dW = -kv.vdt = -kv^2dt

  

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