On a spacecraft two engines fire for a time of 748 s. One gives the craft an acceleration in the x direction of ax = 5.21 m/s2, while the other produces an acceleration in the y direction of ay = 3.45 m/s2. At the end of the firing period, the craft has velocity components of vx = 5390 m/s and vy = 3470 m/s. Find (a) the magnitude and (b) the direction of the initial velocity. Express the direction as an angle with respect to the +x axis.

To solve this problem, we can use the kinematic equations to find the initial velocity components and then use trigonometry to find the magnitude and direction of the initial velocity.

Let's start by finding the initial velocity components.

Using the kinematic equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can find the initial velocities in each direction.

In the x direction:
vx = ux + ax*t
5390 m/s = ux + 5.21 m/s^2 * 748 s
5390 m/s = ux + 3894.08 m/s

Rearranging the equation, we get:
ux = 5390 m/s - 3894.08 m/s
ux = 1495.92 m/s

In the y direction:
vy = uy + ay*t
3470 m/s = uy + 3.45 m/s^2 * 748 s
3470 m/s = uy + 2576.6 m/s

Rearranging the equation, we get:
uy = 3470 m/s - 2576.6 m/s
uy = 893.4 m/s

Now that we have the initial velocity components (ux = 1495.92 m/s and uy = 893.4 m/s), we can find the magnitude and direction of the initial velocity.

(a) The magnitude of the initial velocity is given by the Pythagorean theorem:

|u| = sqrt(ux^2 + uy^2)
|u| = sqrt((1495.92 m/s)^2 + (893.4 m/s)^2)
|u| = sqrt(2239925.34 m^2/s^2 + 798222.36 m^2/s^2)
|u| = sqrt(3038147.7 m^2/s^2)
|u| = 1743.90 m/s

Therefore, the magnitude of the initial velocity is approximately 1743.90 m/s.

(b) The direction of the initial velocity can be found using trigonometry. We can use the arctan function to find the angle with respect to the +x axis:

θ = arctan(uy / ux)
θ = arctan(893.4 m/s / 1495.92 m/s)
θ = arctan(0.5973)

Using a calculator, we find:
θ ≈ 30.26 degrees

Therefore, the direction of the initial velocity is approximately 30.26 degrees with respect to the +x axis.