Estimate the speed of the parachutist descending at the constant velocity with open parachute. Assume the area of the parachute, A = 30 m^2, mass of a person, m = 100 kg, and density of air, roh = 1 kg/m^3. Enter the answer limited to the second decimal place.
Hint 1: air pushing on the parachute exerts the resistive force which matches (equal and opposite) the gravitational force on the person. Hint 2: speed of the air striking the parachute is equal to the speed of the parachutist. Hint 3: force exerted by the air on parachute = (impulse exerted by air)/time interval during which the air strikes the parachute Hint 4: mass of air molecules striking the parachute can be found as mass of the air molecules in the cylinder of air below the parachute Hint 5: height of this cylinder = velocity of air multiplied by the time interval during which it is stopped by the parachute
I find the hints confusing. Here is the formula you need.
M*g = (1/2)*rho*A*Cd*V^2
Solve for the limiting velocity, V
*Cd is the dimensionless "drag coefficient", which is about 1.5 for a hemispherical parachute shape.
*rho is the density of air
*A is the projected area of the parachute (pi R^2), (not the surface area)
*g is the acceleration of gravity
* M is the mass
For more about the formula, see
http://my.execpc.com/~culp/rockets/descent.html
To estimate the speed of the parachutist descending at a constant velocity with an open parachute, we need to find the balance between the resistive force exerted by the air on the parachute and the gravitational force acting on the person.
Hint 1 tells us that the resistive force exerted by the air on the parachute matches the gravitational force on the person. Therefore, we can equate these forces:
F_resistive = F_gravitational
Hint 2 tells us that the speed of the air striking the parachute is equal to the speed of the parachutist. This means we need to find the speed of the air striking the parachute.
Hint 3 states that the force exerted by the air on the parachute can be found as the impulse exerted by the air divided by the time interval during which the air strikes the parachute. We can express this as:
F_resistive = (impulse exerted by air) / (time interval)
Hint 4 indicates that we can find the mass of air molecules striking the parachute by considering the mass of the air molecules in the cylinder of air below the parachute. We can find the height of this cylinder by multiplying the velocity of the air by the time interval during which it is stopped by the parachute.
Hint 5 gives us the relationship between the height of the cylinder and the velocity of the air, which we can use to solve for the velocity.
Let's calculate the speed of the parachutist step by step:
Step 1: Calculate the gravitational force on the person.
F_gravitational = m * g
where m is the mass of the person (100 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 2: Calculate the force exerted by the air on the parachute.
F_resistive = F_gravitational
Step 3: Calculate the impulse exerted by the air.
Impulse = F_resistive * time interval
Step 4: Calculate the mass of air molecules striking the parachute.
Mass of air molecules = density of air * (area of parachute * height of cylinder)
Step 5: Solve for the velocity of the air (speed of the parachutist).
Velocity of air = height of cylinder / time interval
Once you have the velocity of the air, that will be the speed of the parachutist.