An airplane traveling north at 400 m/s is accelerated due east at a rate of
50 m/s2 for 6 s. If the effects of air resistance and gravity are ignored, what is
the final speed of the plane?
50 m/s^2 * 6s = 300 m/s
v = 400j + 300i
speed = 500
To find the final speed of the plane, we can use the concept of vector addition. Since the airplane is traveling north at a constant speed, we can represent its initial velocity as a vector pointing straight up:
Initial velocity (v1) = 400 m/s, pointing north
The acceleration due east can be represented as a vector pointing to the right:
Acceleration (a) = 50 m/s^2, pointing east
To find the final velocity, we need to add the change in velocity to the initial velocity:
Final velocity (v2) = v1 + Δv
Now, we can calculate the change in velocity (Δv) using the formula:
Δv = a × t
where t is the time taken for acceleration, which is 6 seconds.
Δv = 50 m/s^2 × 6 s
Δv = 300 m/s, pointing east
Adding the change in velocity to the initial velocity gives us the final velocity:
Final velocity (v2) = 400 m/s (north) + 300 m/s (east)
To add these vectors, we can use the Pythagorean theorem because they are perpendicular to each other:
v2 = √[(400 m/s)^2 + (300 m/s)^2]
Calculating this:
v2 = √[(160,000 m^2/s^2) + (90,000 m^2/s^2)]
v2 = √[250,000 m^2/s^2]
v2 = 500 m/s
Therefore, the final speed of the plane is 500 m/s.
To solve this problem, we can use the principles of vector addition. We will break down the airplane's velocity into two components: one in the north direction, and one in the east direction.
Given:
Initial velocity in the north direction (Vnorth) = 400 m/s
Acceleration in the east direction (Aeast) = 50 m/s^2
Time (t) = 6 seconds
First, we need to calculate the change in velocity in the east direction (ΔVeast) using the formula:
ΔVeast = Aeast * t
ΔVeast = 50 m/s^2 * 6 s
ΔVeast = 300 m/s
Next, we can use the Pythagorean theorem to find the magnitude of the final velocity (Vfinal), combining the components in the north and east direction:
Vfinal = √(Vnorth^2 + ΔVeast^2)
Vfinal = √(400 m/s)^2 + (300 m/s)^2
Vfinal = √(160000 m^2/s^2 + 90000 m^2/s^2)
Vfinal = √(250000 m^2/s^2)
Vfinal = 500 m/s
Therefore, the final speed of the plane is 500 m/s.