The horizontal velocity is constant. (Ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed v in a direction that makes an angle α with the horizontal, then the initial vertical velocity vv and the horizontal velocity vh are given by

Question

The horizontal velocity is constant. (Ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed v in a direction that makes an angle α with the horizontal, then the initial vertical velocity vv and the horizontal velocity vh are given by

vv = vsinα and vh = vcosα

You shoot a rifle at an angle of 42 degrees. The bullet leaves your rifle at a height of 6 feet and a speed of 810 feet per second.

It hits the ground after how many seconds at a distance of how many feet?

Answer 1

Recall that the vertical height h is

h(t) = 6 + (810 sin42)t - 16t^2

so, just solve for t when h=0.

Answer 2

To answer this question, we can use the equations for motion in the x and y direction.

First, let's find the initial vertical velocity (vv) and the horizontal velocity (vh) using the given angle (α) and speed (v).

Given:
Angle (α) = 42 degrees
Speed (v) = 810 feet per second

Using the equations:
vv = vsinα
vh = vcosα

Substituting the values:
vv = 810 * sin(42)
vv ≈ 810 * 0.6691
vv ≈ 542.51 feet per second

vh = 810 * cos(42)
vh ≈ 810 * 0.7431
vh ≈ 601.62 feet per second

Now, let's find the time it takes for the bullet to hit the ground. Since we are ignoring air resistance, we can use the equation for the vertical motion:

y = vv * t + (1/2) * g * t^2

Given:
Initial height (y) = 6 feet
Vertical velocity (vv) = 542.51 feet per second
Acceleration due to gravity (g) = 32.17 feet per second squared (assuming we are on Earth)

The equation becomes:
6 = 542.51 * t + (1/2) * 32.17 * t^2

Simplifying the equation:
16.085t^2 + 542.51t - 6 = 0

Now we can solve this quadratic equation to find the time it takes for the bullet to hit the ground. Using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

where a = 16.085, b = 542.51, and c = -6.

Using the formula, we can substitute the values and calculate the time.

Once we have the time, we can find the distance traveled by the bullet horizontally using the equation:

Distance = vh * t

Given:
Horizontal velocity (vh) = 601.62 feet per second
Time (t) = calculated from the previous step

Substituting the values, we can calculate the distance.