A dart gun is fired while being held horizontally at a height of 0.90 m above ground level and while it is at rest relative to the ground. The dart from the gun travels a horizontal distance of 4.00 m. A college student holds the same gun in a horizontal position while sliding down a 45.0° incline at a constant speed of 2.00 m/s. How far will the dart travel if the student fires the gun when it is 1.00 m above the ground?

On the first part, determine the horzontal velocity.

On the second, determine the vertical and horizontal components of the student, and add those to the dart velocity.

To find out how far the dart will travel if the student fires the gun when it is 1.00 m above the ground, we can use the principles of projectile motion.

First, let's break down the problem into two components: the horizontal motion and the vertical motion.

1. Horizontal Motion:
The dart gun is held horizontally, so the initial velocity in the horizontal direction (Vx) is constant and equal to the speed at which the student is sliding down the incline, which is 2.00 m/s.

2. Vertical Motion:
The dart is fired when the gun is 1.00 m above the ground. The initial vertical velocity (Vy) is 0 because the gun is at rest relative to the ground. The dart will only be affected by the force of gravity, causing it to accelerate downward.

Now, let's calculate the time (t) it takes for the dart to hit the ground:

We can use the equation for vertical motion:
y = y0 + Vyt + (1/2)gt^2

Where:
y = final position (0 m, since it hits the ground)
y0 = initial position (1.00 m)
Vy = vertical velocity (0 m/s)
g = acceleration due to gravity (-9.8 m/s^2, assuming downward direction)

0 = 1.00 + (0)t + (1/2)(-9.8)t^2

Rearranging the equation gives us:
4.9t^2 = 1.00

Solving for t gives us:
t = sqrt(1.00/4.9)
t ≈ 0.45 s

Now that we know the time it takes for the dart to hit the ground, we can calculate the horizontal distance (D) traveled by the dart using the horizontal velocity (Vx) and the time (t):

D = Vx * t
D = 2.00 m/s * 0.45 s
D ≈ 0.90 m

Therefore, the dart will travel approximately 0.90 meters if the student fires the gun when it is 1.00 m above the ground.