A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by (a) What are and, the x - and y-components of the velocity of the car as functions of time? (b) What are the magnitude and direction of the velocity of the car at? (b) What are the magnitude and direction of the acceleration of the car at? calculus based

a) ax = (-0.036a) t m/s^3; ay = 0.55 m/s^2

b) |v| = 7.469 m/s; Θ = 59°

c) |a| = 0.62 m/s^2; Θ = 298°

Explanation:

a)

the instantaneous acceleration equation will be equal to:

a = dv/dt

ax (t) = (d * (5-0.018*t^2)) / dt = - 0.036*t

ay (t) = (d * (2+0.55*t)) / dt = 0.55

a = (-0.036*t) i + 0.55j

b)

at a time of 8 seconds, the speed of the vector will be equal to:

v = (5 - 0.018*8^2) i + (2 + 0.55*8) j = 3.85i + 6.4j

the magnitude of the vector will be equal to:

|v| = (vx^2 + vy^2) ^1/2 = (3.85^2 + 6.4^2) ^1/2 = 7.469 m/s

the direction of the vector is equal to:

Θ = tan-1 (vy/vx) = tan-1 (6.4/3.85) = 59°

c)

the vector acceleration at a time of 8 seconds will be equal to:

a = (-0.036*8) i + 0.55j = - 0.288i + 0.55j

the magnitude will be equal to:

|a| = (ax^2 + ay^2) ^1/2 = ((-0.288^2) + 0.55^2) ^1/2 = 0.62 m/s^2

the direction:

Θ = tan-1 (ay/ax) = tan-1 (0.55/-0.288) = 298°