Final answer:
To find the speed at which the fluid leaves the needle, the volume flow rate is calculated and then divided by the needle's cross-sectional area, by using the formula for the volume flow rate (Q = A * v) in fluid dynamics.
Step-by-step explanation:
To determine the speed of the fluid as it leaves the needle when a patient is injected with medicine, we use the principle of fluid dynamics within the scope of physics. In this scenario, we can use the formula for the volume flow rate (Q) which relates to the cross-sectional area of the needle (A) and the velocity of the liquid (v). The formula for the area of a circle is πr^2, where 'r' is the radius. Since we have the inside diameter of the needle, we can find the radius by dividing the diameter by two.
To calculate the speed of the fluid, we use the following steps:
- Convert the inside diameter of the needle from millimeters to meters: 0.114 mm = 0.000114 meters.
- Calculate the radius, which is half of the diameter: 0.000114 meters / 2 = 0.000057 meters.
- Calculate the cross-sectional area of the needle (A) using the area formula for a circle, πr^2.
- Determine the volume flow rate by dividing the volume of medicine injected (2.5 ml, which is 2.5 x 10^-6 m³) by the time (0.65 seconds).
- Finally, calculate the velocity (v) of the fluid by dividing the volume flow rate (Q) by the cross-sectional area (A).
Let's put numbers into the steps:
- The area of the needle's cross-section, A = π * (0.000057 m)^2
- The volume flow rate, Q = 2.5 ml / 0.65 s = 2.5 x 10^-6 m³ / 0.65 s = 3.846 x 10^-6 m³/s
- The velocity, v = Q / A
After performing these calculations, you will get the velocity of the fluid exiting the needle.