A dart gun is fired while being held horizontally at a height of 1.00 m above ground level and while it is at rest relative to the ground. The dart from the gun travels a horizontal distance of 5.00 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?
To solve this problem, we need to analyze the motion of the dart in both horizontal and vertical directions. We will assume that there is no air resistance.
First, let's analyze the horizontal motion of the dart. Since the dart gun is being held horizontally, the initial horizontal velocity of the dart is the same as the horizontal velocity of the student, which is 2.00 m/s.
Now, let's analyze the vertical motion of the dart. The dart is fired from a height of 1.00 m above the ground, and it will fall freely under the influence of gravity while traveling horizontally. The vertical distance the dart falls can be determined using the formula:
h = (1/2) * g * t^2
Where h is the vertical distance, g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken. Since the dart is fired horizontally, the time taken to reach the ground is the same as the time taken for the horizontal distance. We can use the horizontal distance formula to find the time taken:
d = v * t
Where d is the horizontal distance, v is the horizontal velocity, and t is the time taken. Rearranging the formula, we have:
t = d / v
Substituting the values, we get:
t = 5.00 m / 2.00 m/s = 2.50 s
Now, let's plug this value into the vertical distance formula:
h = (1/2) * g * t^2
h = (1/2) * 9.8 m/s^2 * (2.50 s)^2 = 30.625 m
Therefore, the dart falls 30.625 m vertically during the time it takes to travel horizontally.
To determine the total distance the dart travels, we can use the Pythagorean theorem, which states that the square of the hypotenuse (total distance) of a right triangle is equal to the sum of the squares of the other two sides (horizontal and vertical distances).
Using this theorem, we can find the total distance:
total distance^2 = horizontal distance^2 + vertical distance^2
total distance^2 = (5.00 m)^2 + (30.625 m)^2
total distance^2 = 25 m^2 + 939.06 m^2
total distance^2 = 964.06 m^2
Taking the square root of both sides, we find:
total distance ≈ 31.07 m
Therefore, the dart will travel approximately 31.07 m if the student fires the gun when it is 1.00 m above the ground.