A 3.9 kg body is initially moving northward at 15 m/s. Then a force of 13 N, toward the east, acts on it for a time of 3.8 s. At the end of that time, what is the body's final velocity? (magnitude and direction)

To find the body's final velocity, we need to calculate the vector sum of its initial velocity and the change in velocity caused by the force acting on it.

Step 1: Determine the initial velocity components.
The initial velocity of the body consists of a northward component and no component in the eastward direction. Given that the body is moving northward at 15 m/s, the initial velocity components are:
Vx(initial) = 0 m/s (eastward component)
Vy(initial) = 15 m/s (northward component)

Step 2: Determine the change in velocity caused by the force.
The force acts only in the eastward direction, so it will only change the eastward component of the velocity. The mass of the body is given as 3.9 kg, and the force acting on it is 13 N. We can use Newton's second law (F = ma) to determine the change in velocity. Rearranging the formula, we have:
a = F / m = 13 N / 3.9 kg ≈ 3.33 m/s^2

Since the force is applied for a time of 3.8 s, we can use the formula for calculating the change in velocity (Δv = at):
Δv = a * t = 3.33 m/s^2 * 3.8 s ≈ 12.67 m/s (eastward component)

Step 3: Calculate the final velocity components.
To find the final velocity, we need to add the change in velocity to the initial velocity. Adding the eastward components and northward components separately, we get:
Vx(final) = Vx(initial) + Δv = 0 m/s + 12.67 m/s ≈ 12.67 m/s (eastward component)
Vy(final) = Vy(initial) = 15 m/s (northward component)

Step 4: Calculate the magnitude and direction of the final velocity.
To find the magnitude of the final velocity, we can use the Pythagorean theorem:
Magnitude of the final velocity (V(final)) = √(Vx(final)^2 + Vy(final)^2) ≈ √(12.67 m/s)^2 + (15 m/s)^2 ≈ √244.92 m^2/s^2 ≈ 15.64 m/s
(Note: The magnitude of the final velocity is rounded to two decimal places.)

To determine the direction of the final velocity, we can use trigonometry. The direction can be found using the angle θ, where θ = atan(Vy(final) / Vx(final)):
θ = atan(15 m/s / 12.67 m/s) ≈ atan(1.18) ≈ 49.32 degrees

Therefore, the body's final velocity is approximately 15.64 m/s at an angle of 49.32 degrees east of north.