A gun with a muzzle velocity of 1290 feet per second is fired at an angle of 8° above the horizontal. Find the vertical and horizontal components of the velocity. (Round ans to one decimal place)

Answer 1

The gun was fired at 1290 feet/sec; so, to begin, let's draw a line showing this initial velocity. The length of the line on the paper isn't important, as long as we label it 1290 feet/sec. Since the gun was pointed upwards 8 degrees, the line shouldn't be horizontal, but rather upward at an 8 degree angle. This line is called a vector, because it has both a magnitude (1290 feet/sec) and direction (8 degrees above horizontal).

A vector can be split into components by drawing a second horizontal line and a third vertical line connecting the vector's tip to tail. These three lines together form a right triangle. Using trigonometry, we can find the length of the vertical line (vertical component of the velocity). The SOH in the acronym SOHCAHTOA reminds us that taking the sine of any angle on a right triangle is equal to the opposite side (side directly across from the chosen angle) divided by the hypotenuse:

SIN(8) = OPPOSITE / 1290

Simplified, this would become:

OPPOSITE = 1290 * SIN(8) = 179.5

So the vertical component of the velocity is 179.5 feet/sec. This means that the bullet is traveling with a speed of 179 feet/sec in the up direction.

--

The length of the horizontal line can be found similarly. The CAH in the acronym SOHCAHTOA tells us that taking the cosine of any angle in a right triangle equals the adjacent side (next to the chosen angle) divided by the hypotenuse:

COS(8) = ADJACENT / 1290

Simplified, this would become:

ADJACENT = 1290 * COS(8) = 1277.4

The horizontal component of the velocity is 1277.4 feet/sec. This means that the bullet is traveling with a speed of 1277 feet/sec in the sideways direction. Remember, we found the vertical component of the velocity was 179.5 feet/sec. Together, these make the total initial velocity 1290 feet/sec in the up/sideways direction of 8 degrees.

--

Note: When doing calculations with SIN and COS, make sure that the calculator is in the correct angle mode. Since we were given our angle in degrees, we need to put the calculator in "Degree Mode" before entering the numbers in. The other option is "Radian Mode", which is commonly used in math courses like algebra 2, precalculus, and calculus.

Answer 2

We use the following to get the vertical (v_y) and horizontal (v_x) components of velocities,

v_x=v*cos θ

v_y=v*sin θ

where v = 1290 ft/s and θ = 8 degrees. Make sure your calculator is set to 'degrees', not 'radians'.

Plug in the numbers and the components become

v_x=1290*cos 8 = 1277 ft/s

v_y=1290*sin 8 = 179.5 ft/s