A dart gun is fired while being held horizontally at a height of 1.19 m above ground level, and at rest relative to the ground. The dart from the gun travels a horizontal distance of 3.33 m. A child holds the same gun in a horizontal position while sliding down a
36.4 degree incline at a constant speed of 2.3 m/s. What horizontal distance x will the dart
travel if the child fires the gun forward when
it is 0.731 m above the ground? The acceleration due to gravity is 9.8 m/s^2.
To solve this problem, we can apply the principles of projectile motion.
Let's denote:
- h1 as the initial height of the dart gun when fired horizontally (1.19 m)
- d1 as the horizontal distance traveled by the dart when fired horizontally (3.33 m)
- h2 as the initial height of the dart gun when fired by the child (0.731 m)
We need to find the horizontal distance traveled by the dart when fired by the child, denoted as x.
First, let's determine the initial velocity of the dart when it is fired by the child. The velocity is the same in the horizontal direction, but we need to find the vertical component of the velocity.
The child is sliding down the incline with a constant speed of 2.3 m/s. The vertical component of this velocity is given by:
v_vertical_child = 2.3 m/s * sin(36.4°)
Now, we can determine the horizontal velocity of the dart when it is fired by the child. This velocity will be the same as the child's horizontal velocity:
v_horizontal_child = 2.3 m/s * cos(36.4°)
Now, we can calculate the time it takes for the dart to reach its maximum height.
Using the equation:
h = h0 + v0 * t - (1/2) * g * t^2
where h is the height, h0 is the initial height, v0 is the initial vertical velocity, t is the time, and g is the acceleration due to gravity.
At the maximum height, the vertical velocity becomes 0, so the equation becomes:
0 = h0 + v0 * t - (1/2) * g * t^2
Simplifying the equation, we get:
t^2 - (2v0/g) * t + (2h0/g) = 0
Using the quadratic formula, we can solve for t:
t = [-(2v0/g) ± √((2v0/g)^2 - 4(2/g)h0)] / 2
Substituting the values:
v0 = v_vertical_child,
g = 9.8 m/s^2,
h0 = h2 (initial height when fired by the child),
we can calculate the time t.
Now that we have the time it takes for the dart to reach its maximum height, we can calculate the horizontal distance traveled by the dart when fired by the child using the equation:
x = v_horizontal_child * t
Substituting the values, we can calculate the horizontal distance x, which is the answer to the question.
To solve this problem, we first need to determine the time it takes for the dart to reach the ground when fired horizontally at a height of 1.19 m.
We can use the equation for the vertical motion of an object in free fall:
y = y₀ + v₀y * t + 1/2 * a * t²
Where:
- y is the final vertical position (0 m, since it reaches the ground)
- y₀ is the initial vertical position (1.19 m)
- v₀y is the initial vertical velocity (0 m/s, since the dart is fired horizontally)
- a is the acceleration due to gravity (-9.8 m/s²)
- t is the time it takes to reach the ground (what we want to find)
Substituting the known values into the equation:
0 = 1.19 + 0 * t + 1/2 * (-9.8) * t²
0 = 1.19 - 4.9t²
Rearranging the equation:
4.9t² = 1.19
t² = 1.19 / 4.9
t² = 0.243
t = √0.243
t ≈ 0.493 s
Now we can move on to finding the horizontal distance the dart will travel when the child fires it at a height of 0.731 m above the ground.
First, we need to find the initial horizontal velocity of the gun when it is fired. We know that the child is sliding down the incline at a constant speed of 2.3 m/s. The horizontal component of this velocity will be the same as the initial horizontal velocity of the dart.
The initial horizontal velocity can be found using the equation:
v₀x = v * cos(θ)
Where:
- v₀x is the initial horizontal velocity
- v is the speed of the child (2.3 m/s)
- θ is the angle of the incline (36.4°)
Substituting the known values into the equation:
v₀x = 2.3 m/s * cos(36.4°)
v₀x ≈ 1.86 m/s
Now, we can find the horizontal distance traveled by the dart using the equation for horizontal motion:
x = v₀x * t
Substituting the known values into the equation:
x = 1.86 m/s * 0.493 s
x ≈ 0.916 m
Therefore, the dart will travel approximately 0.916 meters horizontally when the child fires the gun at a height of 0.731 meters above the ground.