A dart gun is fired while being held horizontally at a height of 1.38m above ground level, and at rest relative to the ground. The dart from the fun travels a horizontal distance of 7.76. A child holds the same gun in a horizontal position while sliding down a 28.8 degrees incline at a constant speed of 1.78 m/s. (gravity is 9.8 m/ssquared)

What horizontal distance x will the dart travel if the child fires the gun forward when it is 0.561 m above the ground?

The first bits of information can be used to determine the muzzle velocity of the gun. Call that velocity Vm. Let T be the time it takes to hit the ground when fired horizontally.

7.76 m = Vm*T
(1/2)gT^2 = 1.38 m
T = 0.531 s
Vm = 14.6 m/s

When sliding down the 28.8 degree incline with the gun aimed horizontaly, the initial velocity components of the dart, in laboratory cordinates, are:
Vx = Vm + 1.78 cos28.9 = 16.2 m/s
Vy = -1.78 sin28.9 = -0.860 m/s
Y' (when fired) = 0.561 m
X' (when fired) = 0

Solve for X' when Y' = 0

You do the rest.

To find the horizontal distance traveled by the dart when the child fires the gun, we can use the principles of projectile motion. We need to split the motion into vertical and horizontal components.

Step 1: Find the vertical component of the dart's velocity when the child fires the gun.
The dart is fired when it is 0.561 m above the ground. The vertical velocity of the dart can be found using the equation:

v_y = v₀_y + a_y * t

Where:
v_y is the vertical velocity
v₀_y is the initial vertical velocity
a_y is the acceleration due to gravity (-9.8 m/s²)
t is the time taken for the dart to reach the given height

First, calculate the time taken for the dart to reach a height of 0.561 m using the formula:

Δy = v₀_y * t + (1/2) * a_y * t²

Rearrange the equation:

0.561 m = 0 + (1/2) * (-9.8 m/s²) * t²

Solve for t:

t² = (2 * 0.561 m) / (9.8 m/s²)
t² = 0.11449 s²
t = √(0.11449 s²)
t ≈ 0.3385 s (approx.)

Now substitute the value of t into the equation to find v_y:

v_y = 0 + (-9.8 m/s²) * 0.3385 s
v_y ≈ -3.32 m/s (approx.)

Note: The negative sign indicates that the velocity is directed downward.

Step 2: Find the horizontal component of the dart's velocity.
Since the child is sliding down the incline at a constant speed, the horizontal component of their velocity will be the same as the dart's horizontal velocity. We can use the equation:

v_x = v₀_x

Where:
v_x is the horizontal velocity
v₀_x is the initial horizontal velocity

The speed of the child sliding down the incline is 1.78 m/s. The initial horizontal velocity can be found using the equation:

v₀_x = v_child * cos(θ)

Where:
v_child is the speed of the child sliding down the incline
θ is the angle of the incline (28.8 degrees)

Convert the angle from degrees to radians:

θ_rad = 28.8 × (π/180) rad
θ_rad ≈ 0.5026 rad (approx.)

Substitute the values into the equation:

v₀_x = 1.78 m/s * cos(0.5026 rad)
v₀_x ≈ 1.485 m/s (approx.)

Step 3: Find the horizontal distance traveled by the dart.
To find the horizontal distance traveled by the dart, we can use the equation:

x = v_x * t

Substitute the values we found into the equation:

x = 1.485 m/s * 0.3385 s
x ≈ 0.5022 m (approx.)

Therefore, the horizontal distance x that the dart will travel when the child fires the gun is approximately 0.5022 meters.