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?
see above.
To determine how far the dart will travel when the student fires the gun, we need to find the horizontal distance covered by the dart.
Let's break down the problem into two parts:
1. Determining the time it takes for the dart to hit the ground.
2. Calculating the horizontal distance traveled by the dart during that time.
Part 1: Determining the time:
We will use the vertical motion equation: h = ut + (1/2)gt^2, where
- h is the vertical distance traveled by the dart (given as 0.90 m),
- u is the initial vertical velocity of the dart (0 m/s, as it is at rest),
- g is the acceleration due to gravity (-9.8 m/s^2),
- t is the time taken.
Rearranging the equation, we get: t = sqrt((2h)/g)
Substituting the values: t = sqrt((2 * 0.90) / 9.8) = 0.429 s (approximately)
Part 2: Calculating the horizontal distance:
Since the student is sliding down the incline at a constant speed of 2.00 m/s, the horizontal velocity (Vx) of the dart will also be 2.00 m/s.
Now, we can use the formula: distance = velocity × time
Substituting values, distance = 2.00 m/s × 0.429 s
distance ≈ 0.858 m
Therefore, the dart will travel approximately 0.858 meters when the student fires the gun at a height of 1.00 m above the ground.