a gun with a muzzle velocity of 1200 feet per second is fired at an angle of 6 degrees above the horizontal. find the component form of a vector. verify the magnitude
Vx = 1200 cos 6
Vy = 1200 sin 6
|V| = sqrt (Vx^2 + Vy^2)
To find the component form of a vector, we need to break down the vector into its horizontal and vertical components. Given that the muzzle velocity of the gun is 1200 feet per second and it is fired at an angle of 6 degrees above the horizontal, we can calculate the horizontal and vertical components as follows:
Horizontal Component:
The horizontal component represents the velocity in the x-axis direction. To find it, we can use the formula: horizontal component = velocity * cos(angle).
horizontal component = 1200 ft/s * cos(6 degrees)
Vertical Component:
The vertical component represents the velocity in the y-axis direction. To find it, we can use the formula: vertical component = velocity * sin(angle).
vertical component = 1200 ft/s * sin(6 degrees)
Now we have the horizontal and vertical components of the vector. Let's calculate them:
horizontal component = 1200 ft/s * cos(6 degrees) ≈ 1196.3 ft/s
vertical component = 1200 ft/s * sin(6 degrees) ≈ 102.2 ft/s
Therefore, the component form of the vector is (1196.3 ft/s, 102.2 ft/s).
To verify the magnitude of the vector, we can use the Pythagorean theorem, which states that the magnitude of a vector is given by the square root of the sum of the squares of its components:
magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
magnitude = sqrt((1196.3 ft/s)^2 + (102.2 ft/s)^2) ≈ 1200.0 ft/s
The magnitude of the vector is approximately equal to 1200.0 ft/s, which verifies our calculations.