Calculate the velocity, speed, tangential and normal accelerations of a moving object whose trajectory is r(t) = [ct cos t; ct sin t; ct]; c ̸= 0.
v= r’(t) = <c.cos(t) – ct.sin(t), c.sin(t) + ct.cos(t), c> ----------------------------------------- (1)
speed = |v| = √{c^2.cos^2(t) +c^2t^2.sin^2(t) – 2c^2t.cos(t).sin(t)
+ c^2.sin^2(t) + c^2t^2.cos^2(t) + 2c^2t.sin(t).cos(t)
+ c^2}
Speed = √{c^2 + c^2t^2 + c^2}
Speed = √{2c^2 + c^2t^2} = c.√{2 + t^2}
Speed = c.√{2 + t^2} --------------------------------------------------------------------------------- (2)
Acceleration: a = r‘‘(t) = <-c.sin(t) – c.sin(t) – ct.cos(t),
c.cos(t) + c.cos(t) – ct.sin(t),
0>
a = <-2c.sin(t) – ct.cos(t), 2c.cos(t) – ct.sin(t), 0> ---------------------------------- (3)
Tangential Accln.
aT = d(|v|)/dt = d(c.√{2 + t^2})/dt
aT = ct/√{2 + t^2}
Acceln:
a = aT.T + aN.N, or
|a|^2 = aT^2 + aN^2 ---------------------------------------------------------------------------------- (4)
Using (3), |a|^2 = 4c^2.sin^2(t) + c^2t^2.cos^2(t) + 4c^2t.sin(t).cos(t)
+ 4c^2.cos^2(t) + c^2t^2.sin^2(t) – 4c^2t.cos(t).sin(t)
+ 0
|a|^2 = 4c^2 + c^2t^2
|a| = c.√{4 + t^2}
Rearranging (4), to give normal accln.
aN^2 = |a|^2 – aT^2
aN^2 = c^2{4 + t^2} – c^2t^2/{2 + t^2}
aN^2 = [c^2{4 + t^2}{2 + t^2} – c^2t^2] / {2 + t^2}
aN^2 = [{4c^2 + c^2t^2}{2 + t^2} – c^2t^2] / {2 + t^2}
aN^2 = [8c^2 + 2c^2t^2 + 4c^2t^2 + c^2t^4 – c^2t^2] / {2 + t^2}
aN^2 = [8c^2 + 5c^2t^2 + c^2t^4] / {2 + t^2}
Normal accln: aN = c.√([8 + 5t^2 + t^4] / {2 + t^2})