Harry Potter is caught in a fierce duel between him and his rival Draco Malfoy. Draco just missed Harry with the Expelliarmus spell, and Harry fires back with a Flipendo jinx, attempting to send Draco flying backward. Harry is standing 44 feet away from Draco and shot the spell at an angle of 38 degrees from his wand which is four feet off the ground. The Flipendo jinx travels at a rate of 15 ft/sec. Disarmed, as soon as Harry unleashed the spell Draco started to run in the opposite direction at a rate of 6 ft/sec. Will the spell ever hit Draco before it dies out? How long will the spell travel before it reaches the ground? Assume that gravity is in effect, but there is no air resistance in the Hogwarts castle.
To determine whether the spell will hit Draco before it dies out, we can calculate the time it takes for the spell to reach Draco and compare it to the time it takes for the spell to die out.
First, let's find the initial velocity of the spell in the horizontal direction. We can use the given angle and the speed of the spell: 15 ft/sec.
The initial horizontal velocity (Vx) can be calculated using the formula: Vx = V * cos(angle),
where V is the speed and the angle is given in radians.
Vx = 15 ft/sec * cos(38 degrees)
= 15 ft/sec * cos(38 * (pi/180)) [Converting degrees to radians]
≈ 11.65 ft/sec
Now, let's find the time it takes for the spell to reach Draco. We can use the horizontal distance and the horizontal velocity.
The horizontal distance (x) is given as 44 ft.
Time (t) can be calculated using the formula: x = Vx * t,
where x is the distance and Vx is the horizontal velocity.
44 ft = 11.65 ft/sec * t
t = 44 ft / 11.65 ft/sec
t ≈ 3.78 sec
So, it will take approximately 3.78 seconds for the spell to reach Draco if Draco doesn't move.
However, Draco is running in the opposite direction at a rate of 6 ft/sec. This means that Draco will move 6 ft/sec * 3.78 sec = 22.68 ft while the spell is traveling towards him.
Now, let's calculate the time it takes for the spell to hit Draco, taking his movement into account.
The effective distance between Harry and Draco is 44 ft - 22.68 ft = 21.32 ft.
Time (t_effective) can be calculated using the formula: x = Vx * t_effective.
21.32 ft = 11.65 ft/sec * t_effective
t_effective = 21.32 ft / 11.65 ft/sec
t_effective ≈ 1.83 sec
Since t_effective is less than the time it takes for the spell to die out (3.78 sec), it means that the spell will hit Draco before it dies out.
To calculate the distance the spell will travel before it reaches the ground, we can calculate the vertical distance.
The initial vertical velocity (Vy) can be calculated using the formula: Vy = V * sin(angle),
where V is the speed and the angle is given in radians.
Vy = 15 ft/sec * sin(38 degrees)
= 15 ft/sec * sin(38 * (pi/180)) [Converting degrees to radians]
≈ 9.06 ft/sec
To find the time it takes for the spell to reach the ground, we can use the vertical distance and the vertical velocity.
The vertical distance (y) from the wand to the ground is given as 4 ft.
Time (t_vertical) can be calculated using the formula: y = Vy * t_vertical + (1/2) * g * t_vertical^2,
where y is the distance, Vy is the vertical velocity, g is the acceleration due to gravity (assumed to be 32 ft/sec^2), and t_vertical is the time.
4 ft = 9.06 ft/sec * t_vertical + (1/2) * 32 ft/sec^2 * t_vertical^2
2 ft = 4.53 ft/sec * t_vertical + 16 ft/sec^2 * t_vertical^2
16 ft/sec^2 * t_vertical^2 + 4.53 ft/sec * t_vertical - 2 ft = 0
Using the quadratic formula, we can solve for t_vertical. The equation becomes:
t_vertical = (-4.53 ft/sec ± √(4.53 ft/sec)^2 + 4 * 16 ft/sec^2 * 2 ft) / (2 * 16 ft/sec^2)
t_vertical ≈ 0.084 sec (ignoring the negative value)
Therefore, the spell will travel for approximately 0.084 seconds before it reaches the ground.
In conclusion, the Flipendo jinx spell will hit Draco before it dies out and will travel for approximately 0.084 seconds before it reaches the ground.