A sprinkler mounted on the ground sends out a jet of water at a 35∘ angle to the horizontal. The water leaves the nozzle at a speed of 15m/s . How far does the water travel before it hits the ground?
Recall that the range
R = (v^2 sin2θ)/g
so, just plug in your numbers.
V= 15m/s at 35° Vox = 15cos35 = 12.29 Voy = 15sin35° = 8.60
Ax = 0 bc acceleration is constant Ay = gravity = -9.8
Equation for time to max height is Vfy = Voy + at
0 = 8.6 + (-9.8)t
-8.6 = -9.8t
.8776s = t to max height
.8776 (2) = 1.755s = t
(.8776 was only the time to max height so double for full time.)
Distance traveled / displacement = Δx
Δx = Vox(t)
Δx = 12.29(1.755)
Δx = 21.57m
The water travels 21.6 m before hitting the ground.
0.22
7.85
Why did the water decide to become an acrobat? Because it wanted to show off its "jet" skills! Now, let's calculate how far our water acrobat travels.
To find the horizontal distance the water travels, we need to determine the time it takes for the water to hit the ground. We can start by breaking down the initial velocity into horizontal and vertical components.
The horizontal component of the velocity (Vx) remains constant throughout the motion, while the vertical component of the velocity (Vy) changes due to gravity.
Vy = V * sin(θ)
Vy = 15 m/s * sin(35°)
Vy ≈ 8.59 m/s
Now, we can determine the time it takes for the water to reach the ground using the equation:
-2 * (Vertical displacement) = Vy * t + 0.5 * g * t^2
Since the water starts at ground level, the vertical displacement is zero.
-2 * 0 = 8.59 m/s * t + 0.5 * 9.8 m/s^2 * t^2
Simplifying the equation, we have:
-19.6 t^2 + 8.59 t = 0
To find the time it takes for the water to reach the ground, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case:
a = -19.6
b = 8.59
c = 0
t = (-8.59 ± √(8.59^2 - 4 * -19.6 * 0)) / (2 * -19.6)
t ≈ 0 or t ≈ 0.877 s
Since we are interested in the positive value (time can't be negative), the water takes approximately 0.877 seconds to reach the ground.
Finally, we can calculate the horizontal distance (d) traveled by the water using the equation:
d = Vx * t
d = (15 m/s * cos(35°)) * 0.877 s
d ≈ 12.84 m
Therefore, the water travels approximately 12.84 meters before it hits the ground. Remember, this is just an estimate, as the calculations are based on ideal conditions. And always keep a safe distance, or you might end up feeling like a soggy clown!
To solve this problem, we can use the principles of projectile motion. Here's how:
Step 1: Identify the given information:
- Angle of the water jet: 35 degrees (θ)
- Initial velocity of the water: 15 m/s (v0)
Step 2: Identify the unknown variable:
- Distance traveled by the water before hitting the ground: d
Step 3: Analyze the motion horizontally:
The horizontal motion is not affected by gravity. The initial velocity in the horizontal direction (v0x) remains constant throughout the motion.
Step 4: Calculate the horizontal component of the initial velocity:
The horizontal component of the initial velocity is given by:
v0x = v0 * cos(θ)
v0x = 15 m/s * cos(35°)
Step 5: Analyze the motion vertically:
The vertical motion is affected by gravity. The water's initial velocity in the vertical direction (v0y) decreases continuously due to the downward acceleration of gravity (9.8 m/s^2).
Step 6: Calculate the vertical component of the initial velocity:
The vertical component of the initial velocity is given by:
v0y = v0 * sin(θ)
v0y = 15 m/s * sin(35°)
Step 7: Calculate the time of flight:
The time taken for the water to hit the ground can be calculated using the vertical motion. The vertical displacement (y) is zero when the water hits the ground. The equation for vertical displacement is given by:
y = v0y * t - (1/2) * g * t^2
Since y = 0, we can solve for t:
0 = v0y * t - (1/2) * g * t^2
(1/2) * g * t^2 = v0y * t
t = 2 * v0y / g
Step 8: Calculate the horizontal distance traveled:
The horizontal distance traveled by the water can be calculated using the horizontal motion. The equation for horizontal displacement is given by:
d = v0x * t
Substitute the value of t from step 7 into the equation above to get the value of d.
Step 9: Calculate the final answer:
Plug in the values and calculate:
v0x = 15 m/s * cos(35°)
v0y = 15 m/s * sin(35°)
t = 2 * v0y / g
d = v0x * t
By substituting these values into the respective equations, you can find the final answer.