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 find the horizontal distance x that the dart will travel when the child fires the gun forward, we can use the principle of projectile motion.
First, let's analyze the motion of the dart when it is fired while being held horizontally at a height of 1.19 m above the ground.
1. The initial vertical velocity (Vy) of the dart is 0 m/s because it is at rest relative to the ground.
2. The dart is only affected by the acceleration due to gravity (g = 9.8 m/s^2) in the vertical direction.
3. The time of flight (t) can be found using the formula: t = sqrt(2h/g), where h is the initial vertical height. In this case, h = 1.19 m.
Therefore, t = sqrt(2 * 1.19 / 9.8) = 0.494 s.
4. The horizontal distance (x) traveled by the dart can be calculated using the formula: x = Vx * t, where Vx is the horizontal velocity, and t is the time of flight.
However, in this case, the dart is fired horizontally, so Vx = 0 m/s.
Therefore, x = 0 m.
Therefore, when the dart is fired while being held horizontally, it will not travel any horizontal distance.
Now let's analyze the motion of the dart when the child fires the gun forward while sliding down the incline.
1. The child is sliding down the incline at a constant speed of 2.3 m/s. We can decompose this velocity into its components:
Vertical component (Vyi) = 2.3 * sin(36.4) = 1.36 m/s
Horizontal component (Vxi) = 2.3 * cos(36.4) = 1.85 m/s
2. The initial vertical height (h) at which the gun is fired is given as 0.731 m.
3. The time of flight (t) in this case can be found using the formula: t = (2 * h) / (g + Vyi), where h is the initial vertical height, g is the acceleration due to gravity, and Vyi is the initial vertical component of velocity.
Therefore, t = (2 * 0.731) / (9.8 + 1.36) = 0.144 s.
4. The horizontal distance (x) traveled by the dart can be calculated using the formula: x = Vxi * t, where Vxi is the horizontal component of velocity, and t is the time of flight.
Therefore, x = 1.85 * 0.144 = 0.266 m.
Therefore, when the child fires the gun forward while sliding down the incline, the dart will travel a horizontal distance of 0.266 m.