A Knight of the Round Table fires off a vat of
burning pitch from his catapult at 17.4 m/s,
at 33 ◦
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.
What is the horizontal component of the velocity?
Answer in units of m/s
Why did the Knight of the Round Table fire off a vat of burning pitch? Because he wanted to make a flaming entrance at the jousting tournament, of course!
Now, let's calculate the time the vat of burning pitch is in the air and the horizontal component of its velocity.
To find the time, we can use the formula for projectile motion. The vertical component of the initial velocity will be given by v₀y = v₀ * sin(θ), where v₀ is the initial velocity and θ is the angle above the horizontal. In this case, v₀y = 17.4 m/s * sin(33°).
The time of flight can be determined using the formula t = 2 * v₀y / g, where g is the acceleration due to gravity. Plugging in the values, we get:
t = 2 * (17.4 m/s * sin(33°)) / 9.8 m/s²
Now, let's calculate the horizontal component of the velocity. The horizontal component is given by v₀x = v₀ * cos(θ), where v₀ is the initial velocity and θ is the angle above the horizontal. In this case, v₀x = 17.4 m/s * cos(33°).
So, the time the vat of burning pitch is in the air will be the calculated value of t, and the horizontal component of its velocity will be the calculated value of v₀x.
Now, if only the vat of burning pitch could tell us its own story...
To find the time the vat of burning pitch is in the air, we can use the vertical motion equation:
h = v₀y * t + (1/2) * g * t²
Where:
h = the vertical displacement (which is 0 in this case because the vat will land at the same height)
v₀y = the initial vertical velocity
t = time
g = acceleration due to gravity
Since the vat is fired at an angle of 33 degrees above the horizontal, we need to find the vertical component of the initial velocity. We can use trigonometry to do this.
v₀y = v * sin(θ)
Where:
v = the initial velocity of the vat
θ = the angle of projection
Given that v = 17.4 m/s and θ = 33 degrees, we can calculate the vertical component of the initial velocity:
v₀y = 17.4 m/s * sin(33°) ≈ 9.451 m/s
Substituting this value into the equation for vertical motion:
0 = 9.451 m/s * t + (1/2) * 9.8 m/s² * t²
Simplifying this equation:
0 = 4.725 m/s² * t + 4.9 m/s² * t²
Rearranging the equation and setting it equal to zero:
4.9 m/s² * t² + 4.725 m/s² * t = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
Where:
a = 4.9 m/s²
b = 4.725 m/s²
c = 0
Calculating the values and solving for t:
t = (-4.725 ± √(4.725² - 4 * 4.9 * 0)) / (2 * 4.9)
t = (-4.725 ± √(22.330625)) / (9.8)
t = (-4.725 ± 4.721218) / 9.8
This gives us two possible solutions for t:
t₁ = (-4.725 + 4.721218) / 9.8 ≈ 0.000403 s
t₂ = (-4.725 - 4.721218) / 9.8 ≈ -0.000819 s
Since time cannot be negative in this context, we discard the negative value and conclude that the vat of burning pitch is in the air for approximately 0.000403 seconds.
Now, to find the horizontal component of the velocity, we can use the equation:
v₀x = v * cos(θ)
Given that v = 17.4 m/s and θ = 33 degrees, we can calculate the horizontal component of the initial velocity:
v₀x = 17.4 m/s * cos(33°) ≈ 14.553 m/s
Therefore, the horizontal component of the velocity is approximately 14.553 m/s.
To find the time of flight of the vat of burning pitch, we can use the equation of motion for vertical motion. The equation is:
y = y0 + v0y*t - (1/2) * g * t^2
Where:
y = vertical displacement (in this case, the height above the ground)
y0 = initial vertical position (assumed to be 0 in this case)
v0y = vertical component of the initial velocity (in this case, v0y = v0 * sin(theta))
g = acceleration due to gravity
Let's find the time of flight:
1. Convert the angle from degrees to radians:
theta = 33 degrees * (π/180) = 0.575958653 radians
2. Calculate the vertical component of the initial velocity:
v0y = v0 * sin(theta)
= 17.4 m/s * sin(0.575958653)
= 9.049453800 m/s
3. Use the equation of motion to find the time of flight:
0 = 0 + 9.049453800 * t - (1/2) * 9.8 * t^2
Rearrange the equation to become a quadratic equation:
4.9 * t^2 - 9.049453800 * t = 0
Solve the equation by factoring:
t * (4.9 * t - 9.049453800) = 0
Therefore, t = 0 (This is the initial time, not the time of flight) or t = 1.849868932 seconds.
So, the time of flight of the vat of burning pitch is approximately 1.8499 seconds.
To find the horizontal component of the velocity, we can use the equation:
vx = v0x
Where:
vx = horizontal component of the velocity
v0x = initial horizontal velocity (in this case, v0x = v0 * cos(theta))
Let's find the horizontal component of the velocity:
1. Calculate the horizontal component of the initial velocity:
v0x = v0 * cos(theta)
= 17.4 m/s * cos(0.575958653)
= 15.05199857 m/s
Therefore, the horizontal component of the velocity is approximately 15.052 m/s.
u = horizontal velocity = 17.4 cos 33
Vi = initial vertical velocity = 17.4 sin 33
v = Vi - 9.8 t
at top v = 0
so at top t = Vi/9.8
it spends the samer asmount of time falling from the top so
time in air = 2 t = 2 Vi/9.8