Question

A hypodermic syringe contains a medicine with the density of water. The barrel of the syringe has a cross-sectional area a = 2.50 x 10^-2 m^2, and the needle has a cross-sectional area a = 1.00 x 10^-8 m^2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. A force F of magnitude 2.00 N acts on the plunger, making the medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

Answer 1

The speed of the medicine as it leaves the needle's tip is 17.4 m/s.

Step-by-step explanation:

Bernoulli's equation states that the total energy of a fluid is conserved along a streamline. This equation can be expressed as:

P + 1/2ρv² = constant

where P is the pressure, ρ is the density of the fluid, and v is the velocity of the fluid.

Apply Bernoulli's equation to the fluid in the syringe before and after it exits the needle.

Before exiting the needle:

P = P_atm

v = 0

After exiting the needle:

P = P_needle

v = v

Solve the equation for v:

v = √((2 * (P_needle - P_atm)) / ρ)

where P_atm is the atmospheric pressure, P_needle is the pressure in the needle, and ρ is the density of the fluid.

Plugging in the values, we get:

v = √((2 * (1.013e5 Pa + 2.00 N / 2.50e-2 m^2)) / 1000 kg/m^3)) = 17.4 m/s

Therefore, the speed of the medicine as it leaves the needle's tip is 17.4 m/s.