Final answer:
The horizontal component of the velocity of the pitched vat fired from the catapult is approximately 11.66 m/s. If we assume a level launch and landing, it is in the air for about 1.841 seconds and lands approximately 21.48 meters away from the catapult.
Step-by-step explanation:
A Knight of the Round Table fires off a vat of burning pitch from his catapult at 14.8 m/s, at an angle of 38° above the horizontal. The acceleration due to gravity is 9.8 m/s2. To determine the horizontal component of the velocity, we use the cosine function since it represents the adjacent side of the angle in a right-angled triangle. The formula is Vx = V * cos(θ), where V is the initial velocity and θ is the launch angle. Plugging in the values gives us Vx = 14.8 m/s * cos(38°) = 14.8 m/s * 0.788 = 11.66 m/s.
The time the projectile is in the air, also called the time of flight, and the distance from the catapult it lands can be calculated using the kinematic equations for projectile motion. However, without complete information about the problem, such as the height from where the projectile is launched or lands, we cannot accurately calculate the time in the air or the range of the projectile. If we assume that the launch and landing heights are the same and it's launched and lands at ground level, we can calculate these values.
To determine how long it's in the air, we use the formula for the vertical motion. The time to reach the highest point is t = V * sin(θ)/g. The total time would be twice this value since the upward and downward journey takes equal time. Using the given values, t = (14.8 m/s * sin(38°)) / 9.8 m/s2 = (14.8 m/s * 0.615) / 9.8 m/s2 = 0.9204 s. Therefore, the time in the air is 2 * 0.9204 s = 1.841 s.
The range R of the projectile can be found using R = Vx * t = 11.66 m/s * 1.841 s = 21.48 m. Thus, the vat lands approximately 21.48 meters away from the catapult.