Final answer:
To find the muzzle velocity of the Nerf gun, we need to determine the initial vertical velocity of the foam dart. We can do this by setting the equation for vertical displacement equal to zero and solving for the time it takes for the dart to reach the ground. Then, using the equation for horizontal displacement, we can solve for the initial vertical velocity. Plugging in the given values, we can calculate the muzzle velocity of the Nerf gun.
Step-by-step explanation:
To find the muzzle velocity of the Nerf gun, we need to determine the initial vertical velocity of the foam dart. Since the dart goes off 2 meters above the floor, we can use the equation:
Vertical Displacement = Initial Vertical Velocity * Time + 0.5 * Acceleration * Time^2
Plugging in the values, we get:
2 = Initial Vertical Velocity * Time - 4.9 * Time^2
To solve for the time it takes for the dart to reach the ground, we can set the equation equal to zero:
-4.9 * Time^2 + Initial Vertical Velocity * Time - 2 = 0
Using the quadratic formula, we can solve for Time:
Time = (Initial Vertical Velocity ± sqrt(Initial Vertical Velocity^2 + 4 * 4.9 * 2)) / (2 * -4.9)
Since the foam dart is going up and coming back down, we only need the positive root of the equation. Plugging in the values, we get:
Time = (Initial Vertical Velocity + sqrt(Initial Vertical Velocity^2 + 39.2)) / -9.8
Given that the horizontal displacement of the dart is 11 meters, we can use the equation
Horizontal Displacement = Horizontal Velocity * Time
Plugging in the values, we get:
11 = Horizontal Velocity * Time
We can solve for Time by rearranging the equation:
Time = 11 / Horizontal Velocity
Setting the two equations for Time equal to each other, we can solve for Initial Vertical Velocity:
(Initial Vertical Velocity + sqrt(Initial Vertical Velocity^2 + 39.2)) / -9.8 = 11 / Horizontal Velocity
Multiplying both sides of the equation by -9.8 and rearranging, we get:
Initial Vertical Velocity + sqrt(Initial Vertical Velocity^2 + 39.2) = -11 * Horizontal Velocity / 9.8
Squaring both sides of the equation, we get:
Initial Vertical Velocity^2 + 2 * Initial Vertical Velocity * sqrt(Initial Vertical Velocity^2 + 39.2) + Initial Vertical Velocity^2 + 39.2 = (11 * Horizontal Velocity / 9.8)^2
Combining like terms, we get:
2 * Initial Vertical Velocity^2 + 2 * Initial Vertical Velocity * sqrt(Initial Vertical Velocity^2 + 39.2) + 39.2 - (11 * Horizontal Velocity^2) / (9.8^2) = 0
This is a quadratic equation in terms of Initial Vertical Velocity. Solving for Initial Vertical Velocity will give us the muzzle velocity of the Nerf gun.