An object has a position given by r(t) = (2t^3 + 4t + 2, 4t + 2) m. Answer the f

Question

An object has a position given by r(t) = (2t^3 + 4t + 2, 4t + 2) m. Answer the following: What is its displacement over the interval (1 s, 3 s) in component form? What is its displacement over that interval in magnitude-angle form (Measure the angle from the + x axis) What is its instantaneous velocity at 2 seconds in component form? What is its instantaneous speed at 2 seconds? What is its instantaneous acceleration at 5 seconds in component form? An object has an acceleration given by a(t) = (2, 3) m/s^2. Answer the following: Given that the object's velocity at t = 0 is (-3, -4) m/s, what is its velocity at 5 seconds in component form? What is the object's speed at 5 seconds? What is the object's displacement over the interval (2 s, 3 s) in component form? What is the object's displacement in magnitude-angle form?

Explanation / Answer

1)

i) r = (2t^3 + 4t + 2 , 4t + 2) m

over 1 to 3 s

displacement = ((2* 3^3 + 4*3 + 2) - (2* 1^3 + 4 + 2)) i + ((4* 3 + 2) - (4* 1 + 2)) j

displacement = 60 i + 8 j = (60 , 8) m

ii)

in magnitude angle form

displacement = sqrt(60^2 + 8^2) at arctan(8/60) degree

displacement = 60.53 m at 7.6 degree with + x axis

iii)

for the velocity

v = dr/dt

v = d/dt(2t^3 + 4t + 2 , 4t + 2)

v = (6t^2 + 4 , 4)

at t = 2 s

v = (6 * 4 + 4 , 4)

v = (28 , 4) m/s

iv)

speed at 2 s

speed = sqrt(28^2 + 4^2)

speed = 28.3 m/s

v)

for the acceleration

a = d/dt((6t^2 + 4 , 4))

a = 12 * t

at t = 5 s

a = 60 i + 0 j m/s^2

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