A Knight of the Round Table fires off a vat of

burning pitch from his catapult at 17 m/s, at
29

above the horizontal.
The acceleration of gravity is 9.8 m/s
2
.
How long is it in the air?
Answer in units of s

The initial vertical velocity component Vyo determines the time in the air.

Vyo = 17 sin29 = 8.24 m/s

Vy becomes zero at
t = Voy/g = 0.84 s

Double that time to include the time coming back down.

To find the time it takes for the vat of burning pitch to be in the air, we can use the vertical component of its initial velocity and the acceleration due to gravity.

Given:
Initial velocity (v0) = 17 m/s
Launch angle (θ) = 29 degrees
Acceleration due to gravity (g) = 9.8 m/s^2

First, let's find the vertical component of the initial velocity:
Vy = v0 * sin(θ)
Vy = 17 m/s * sin(29 degrees)
Vy ≈ 8.23 m/s

Using the equation for the vertical motion of an object under constant acceleration:
Vy = v0y + gt

We can rearrange the equation to solve for time (t):
t = (Vy - v0y) / g

Substituting the values:
t = (8.23 m/s - 0 m/s) / 9.8 m/s^2

Calculating:
t ≈ 0.84 seconds

Therefore, the vat of burning pitch will be in the air for approximately 0.84 seconds.

To determine how long the vat of burning pitch is in the air, we can use the kinematic equation for projectile motion. This equation relates the vertical displacement (in this case, the height) to the initial velocity, angle, and time.

The equation we will use is:
Δy = v₀y * t + (1/2) * g * t²

Where:
Δy is the vertical displacement or height (which is 0 in this case, as the vat of pitch is fired at an angle above the horizontal and lands at the same height)
v₀y is the vertical component of the initial velocity (v₀y = v₀ * sin(θ))
t is the time
g is the acceleration due to gravity (g = 9.8 m/s²)

Since the vat of pitch is fired at an angle of 29° above the horizontal, we can calculate the vertical component of the initial velocity (v₀y) using the given initial velocity (v₀ = 17 m/s) and the sine function:
v₀y = v₀ * sin(θ)
v₀y = 17 m/s * sin(29°)

Now, substitute the values into the equation and solve for time (t):

0 = (17 m/s * sin(29°)) * t + (1/2) * (9.8 m/s²) * t²

Rearranging the equation and factoring out t:

0 = t * (17 m/s * sin(29°) + (1/2) * (9.8 m/s²) * t)

Setting the equation equal to zero allows us to solve for t.

Now we can either solve this quadratic equation using the quadratic formula or by factoring.

Using the quadratic formula:
t = (-b ± sqrt(b² - 4ac)) / (2a)

In this case, a = (1/2) * (9.8 m/s²), b = 17 m/s * sin(29°), and c = 0.

Plugging in the values into the formula:

t = (-(17 m/s * sin(29°)) ± sqrt((17 m/s * sin(29°))² - 4 * (0.5 * 9.8 m/s² * 0))) / (2 * 0.5 * 9.8 m/s²)

After calculating, we get two possible solutions for t: a positive value and a negative value. Since time cannot be negative, we will consider the positive value.

Thus, the time the vat of burning pitch is in the air is approximately the positive value of t. Convert this time to the desired units of seconds.