Question

The acceleration function of an object doing curvilinear motion is a = {(-0.2)i+2j+1.5k) m/s², where t is in s. If its initial velocity vo 8i m/s, and initial position is at the origin, determine the magnitude of its velocity when t = 3 s.​

Answer 1

Asnwer:
To solve this problem, we need to integrate the acceleration function to obtain the velocity function, and then evaluate the magnitude of the velocity at t = 3 s.

The velocity function can be obtained by integrating the acceleration function as follows:

v(t) = ∫a(t)dt

v(t) = ∫((-0.2)i+2j+1.5k)dt

v(t) = (-0.2t)i+(2t)j+(1.5t)k + C

where C is a constant of integration. To determine the value of C, we can use the initial velocity, which is given as vo = 8i m/s when t = 0 s:

v(0) = (-0.2(0))i+(2(0))j+(1.5(0))k + C = 8i

C = 8i

Therefore, the velocity function is:

v(t) = (-0.2t)i+(2t)j+(1.5t)k + 8i

Now, to determine the magnitude of the velocity at t = 3 s, we can simply evaluate the velocity function at t = 3 s and compute its magnitude:

v(3) = (-0.2(3))i+(2(3))j+(1.5(3))k + 8i

v(3) = (-0.6)i+6j+4.5k + 8i

v(3) = 7.4i+6j+4.5k

|v(3)| = sqrt((7.4)^2 + 6^2 + (4.5)^2)

|v(3)| = sqrt(102.41 + 36 + 20.25)

|v(3)| = sqrt(158.66)

|v(3)| ≈ 12.6 m/s (rounded to one decimal place)

Therefore, the magnitude of the velocity when t = 3 s is approximately 12.6 m/s.

Answer 2

the magnitude of the velocity when t = 3 s is 10.54 m/s.

Step-by-step explanation:

To solve this problem, we can use the following kinematic equation that relates velocity, acceleration, and time:

v = vo + at

where:

v = final velocity

vo = initial velocity

a = acceleration

t = time

First, we need to find the velocity of the object at time t = 3 s. To do this, we can substitute the given values into the kinematic equation and solve for v:

v = vo + at

v = 8i + (-0.2i+2j+1.5k) x 3

v = 8i - 0.6i + 6j + 4.5k

v = 7.4i + 6j + 4.5k

The magnitude of the velocity is given by:

|v| = sqrt(vx^2 + vy^2 + vz^2)

where:

vx, vy, vz = the x, y, and z components of the velocity vector

Substituting the values from above, we get:

|v| = sqrt((7.4)^2 + 6^2 + (4.5)^2)

|v| = sqrt(54.81 + 36 + 20.25)

|v| = sqrt(111.06)

|v| = 10.54 m/s (approx)