Question

A 77 -kg person is parachuting and experiencing a downward acceleration of \(2.5 \mathrm{~m} / \mathrm{s}^{2}\) shortly after opening the parachute. The mass of the parachute is \(5.2 \mathrm{~kg} .\) (a) Find the upward force exerted on the parachute by the air. (b) Calculate the downward force exerted by the person on the parachute.

Short answer

Upward force exerted on the parachute by the air is 1011.06 N. The downward force exerted by the person on the parachute is 754.6 N.

Step-by-step solution

Calculate total mass and weight

First we calculate the total mass of the system (person + parachute), which is \(m = 77 kg + 5.2 kg = 82.2 kg\). The weight of the system (downward force due to gravity) can be calculated using the formula \(mg\), where \(g = 9.8 m/s^2\) is the acceleration due to gravity. So, the weight is \(F_{gravity} = mg = 82.2 kg × 9.8 m/s^2 = 805.56 N\).

Calculate net force

Next, calculate the net force acting on the system. This is done by multiplying the total mass of the system by the given acceleration. So, the net force is \( F_{net} = m × a = 82.2 kg × 2.5 m/s^2 = 205.5 N\).

Find the upward force on parachute (Part a)

The net force is defined as the difference between the upward force exerted on the parachute by the air and the downward weight of the system. By rearranging the equation for net force: \(F_{net} = F_{upward}-F_{gravity}\), we can solve for the upward force: \(F_{upward} = F_{net} + F_{gravity} = 205.5 N + 805.56 N = 1011.06 N\).

Find the downward force exerted by the person (Part b)

The downward force exerted by the person on the parachute is simply the weight of the person. This can be calculated using the same equation for weight we've used before, but this time only using the mass of the person: \(F_{person} = m_{person} × g = 77 kg × 9.8 m/s^2 = 754.6 N\).

Key concepts

Downward Acceleration

When we discuss downward acceleration, we're talking about how fast something is speeding up as it falls. In physics, this is usually affected by gravity, which on Earth pulls objects down at about 9.8 m/s². In this exercise, the person and the parachute are not falling due to gravity alone. Instead, they experience a downward acceleration of 2.5 m/s², which is less than gravity.
This means the parachute is working against gravity, slowing the person down. When the acceleration is less than the full force of gravity, it's because of the upward force, which helps to decelerate the fall.

Net Force Calculation

Net force is like the tug-of-war result between different forces acting on an object. To find net force, you subtract the total downward forces from the total upward forces.
In the parachuting scenario, net force is the combination of gravity pulling down and the parachute being pulled up by the air. We calculate it by using the formula:

• F_net = total mass × acceleration

The total mass here is the parachuter plus the parachute. Multiplying 82.2 kg by 2.5 m/s² gives 205.5 N, which is the net force propelling the system downward.

Upward Force Calculation

To calculate the upward force exerted on the parachute, we need to consider that the net force we calculated earlier is also the difference between the upward force and the force due to gravity. So, we use the formula:

• F_upward = F_net + F_gravity

The weight of the parachuting person and the parachute system adds up to the gravitational force, which is 805.56 N. Adding this to the net force of 205.5 N, we find that the upward force provided by the air is 1011.06 N.
This upward force is crucial because it reduces the overall speed of the person, allowing for a safe landing.