12-83. Determine the maximum height on the wall to
which the firefighter can project water from the hose, if
the speed of the water at the nozzle is 48ft/s
h = 48t - 16t^2
max height at t=3/2
h(1.5) = ?
To determine the maximum height on the wall to which the firefighter can project water from the hose, we can use the equations of motion for projectile motion.
The initial speed of the water at the nozzle is given as 48 ft/s. We can assume that the water is projected vertically upwards, as this will give us the maximum height.
The equation for the height of a projectile at time t is given by:
h = v0 * t - (1/2) * g * t^2
Where:
- h is the height
- v0 is the initial velocity (in this case, the speed of the water at the nozzle)
- g is the acceleration due to gravity (approximately 32.2ft/s^2)
- t is the time
In this case, we want to find the maximum height, which occurs when the vertical velocity of the water becomes zero. This occurs at the peak of the trajectory, where the time taken to reach the maximum height (t_peak) is half of the total time of flight (t_total).
To find the total time of flight, we first find the time it takes for the water to reach its maximum height (t_peak) using the equation:
0 = v0 - g * t_peak
Solving for t_peak, we get:
t_peak = v0 / g
Then, we can find the total time of flight (t_total) by doubling the time to reach the maximum height:
t_total = 2 * t_peak
Now, we can substitute the values into the equation for the height to find the maximum height:
h_max = v0 * t_peak - (1/2) * g * t_peak^2
Let's calculate the maximum height.
To determine the maximum height the firefighter can project water from the hose, we can use the concepts of projectile motion.
In this case, the water acts as a projectile, moving upwards from the hose nozzle. The initial vertical velocity of the water can be calculated using the speed given, which is 48 ft/s.
First, we need to split the initial velocity into its horizontal and vertical components. Since we are only concerned with the vertical motion, the horizontal component does not affect the final height.
The vertical motion of the water is influenced by the acceleration due to gravity, which is approximately 32.2 ft/s^2. At the highest point of its trajectory, the water reaches its maximum height.
To find the maximum height, we can use the kinematic equation for vertical displacement:
y = (v^2 - u^2) / (2a)
Where:
y = vertical displacement or height
v = final vertical velocity (which is 0 at the maximum height)
u = initial vertical velocity (given as 48 ft/s)
a = acceleration due to gravity (32.2 ft/s^2)
Plugging in the values, we have:
0 = (0 - 48^2) / (2 * -32.2)
Simplifying the equation gives:
0 = -2304 / -64.4
Now, solving for y gives:
y = 2304 / 64.4
y ≈ 35.78 ft
Therefore, the maximum height on the wall to which the firefighter can project water from the hose is approximately 35.78 feet.