A tin can is filled with water to a depth of 30 cm. A hole 11 cm above the bottom of the can produces a stream of water that is directed at an angle of 34° above the horizontal.
(a) Find the range of this stream of water.
(b) Find the maximum height of this stream of water.
Use the Bernoulli equation to get the velocity of the water leaving the hole. There is 0.19 m of water above the hole.
The water pressure there is (rho)*g*h above ambient pressure, which is
1000 kg/m^3*9.81m/s^2*0.19 = 1864 N
(rho is the density of water)
The water flows through the hole at a velocity given by
(1/2)*rho*V^2 = 1864 N
V = 1.93 m/s
Now consider the water in air as a ballistics problem. Drops are launched with a vertical velocity component of 1.93 sin 34 = 1.08 m/s and a horizontal component of 1.60 m/s. The water hits the ground 0.11 m below the hole. Compute the time it takes to hit the ground and multiply that by 1.60 m/s to get the range of the stream.
The maximum height can be computed by computing that the stream reaches maximum height at (1.08 m/s)/g = 0.11 s after leaving the hole. Multiply that by the average vertical velocity component as it reaches maximum height, 0.54 m/s. The stream rises 5.6 cm above the hole, or 16.6 cm above the base.
To find the range of the water stream in this scenario, we can use the concept of projectile motion. The water stream can be treated as a projectile being launched from the hole in the can. Here's how you can find the range:
Step 1: Determine the initial velocity of the water stream.
Since the water is being expelled from a hole, we can assume there is no initial vertical velocity (Vy = 0). The initial horizontal velocity (Vx) can be found using the formula: Vx = V * cosθ, where V is the initial velocity and θ is the launch angle.
Step 2: Calculate the time of flight.
The time it takes for the water stream to reach the ground is known as the time of flight (T). It can be calculated using the formula: T = 2 * Vy / g, where g is the acceleration due to gravity.
Step 3: Find the range.
The range (R) of the water stream is given by the formula: R = Vx * T.
Now let's calculate the range:
Step 1: Determine the initial velocity of the water stream.
In the problem statement, the angle above the horizontal (θ) is given as 34°. We need to find the initial velocity (V). As there is no information about it, we can use the assumption that the water stream leaves the hole vertically downwards. Therefore, the launch angle is 90° - 34° = 56° (measured from the horizontal).
Step 2: Calculate the time of flight.
Since the water stream starts with zero vertical velocity, the time of flight (T) is twice the time it takes for the water stream to reach its maximum height. This can be calculated using the formula: T = 2 * Vy / g, where Vy is the initial vertical velocity (which we assumed to be zero) and g is the acceleration due to gravity, approximately 9.8 m/s².
Step 3: Find the range.
Using the formulas mentioned above, we can now calculate the range: R = Vx * T.
To determine the maximum height of the water stream, we can find the vertical distance it reaches from the starting point (the hole in the can) to the highest point of its trajectory. Here's how you can do it:
Step 1: Determine the initial velocity of the water stream.
In this case, we already have the initial velocity (V) from the previous calculations.
Step 2: Calculate the time it takes to reach the maximum height.
The time it takes for the water stream to reach its maximum height (Tmax) can be calculated using the formula: Tmax = Vy / g.
Step 3: Find the maximum height.
The maximum height (H) of the water stream can be found using the formula: H = Vy * Tmax - 0.5 * g * Tmax².
Now let's calculate the maximum height:
Step 1: Determine the initial velocity of the water stream.
We have already calculated the initial velocity (V) in the previous steps.
Step 2: Calculate the time it takes to reach the maximum height.
Using the formula mentioned above, Tmax = Vy / g.
Step 3: Find the maximum height.
Using the formula mentioned above, H = Vy * Tmax - 0.5 * g * Tmax².
By following these steps and plugging in the appropriate values, you can find both the range and the maximum height of the water stream.