The position of a particle as a function of time is given as x = 12t^2-2.0 t^3,

Question

The position of a particle as a function of time is given as x = 12t^2-2.0 t^3, where x is in meters and t is in seconds, Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at t = 3.0 s. (d) Determine the average velocity of the particle between t = 0 and t = 3.0 s. Given three displacement vectors in meters, d_1 = -3.0i + 3.0j + 2.0k; d_2 = 2.0i - 4.0j + 2.0k; d_3 = 2.0i + 3.0j + 1.0k. Determine the resultant, d1 (d_2xd_3). A shell is fired from the top of an 80-m high cliff with a velocity 20 m/s at an angle 65^degree with the vertical (y axis). A) Determine the horizontal distance (x), measured from the foot of the cliff, where the shell strikes the ground. B) Determine the magnitude of the velocity of the shell as it hits the ground. C) Determine the direction of the velocity of the shell as it hits the ground. D) Draw the trajectory of the shell. A 0.3 kg rubber ball is dropped from the window of a building. it strikes the sidewalk below at 30 m/s and rebounds up at 20 m/s. Determine the magnitude and the direction of the force with which the ball strikes the sidewalk if the collision last for 1 ms. A uniform ball of mass M = 6.00 kg and radius R, rolls smoothly from rest down a ramp at angle e 30^degree. The descends a vertical height h = 1.2 m to reach the bottom of the ramp. Determine the speed of the ball at the bottom. Given, l_com = (2/5)MR^2. Water is moving with a speed of 5.0 m/s through a pipe with a cross-sectional area of 4.0 cm^2. The water gradually descends 10 m as the pipe cross-sectional area increases to 8.0 cm^2. A) What is speed at the lower level? B) If the pressure at the upper level is 1.5 times 10^5 Pa, what is the pressure at the lower level?

Explanation / Answer

Given,

x = 12t^2 - 2 t^3

a)t = 3

x = 12 x 3^2 - 2 x 3^3 = 54 m

Hence, x = 54 m

b)v = d(x)/dt = d(12 t^2 - 2 t^3)

v = 24 t - 6 t^2

t = 3

v = 24 x 3 - 6 x 3^2 = 18 m/s

Hence, v = 18 m/s

c)a = d(v)/dt = d(24 t - 6t^2)

a = 24 - 12 t

t = 3

a = 24 - 12 x 3 = -12 m/s^2

Hence, a = -12 m/s^2

d)for t = 0

v = 0

for t = 3

v = 18

v(av) = (0 + 18)/2 = 9 m/s

Hence, V(av) = 9 m/s

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