Problem 49 - College Physics

Question

A parachutist relies on air resistance (mainly on her parachute) to decrease her downward velocity. She and her parachute have a mass of 55.0 \(\mathrm{kg}\) , and at a particular moment air resistance exerts a total upward force of 620 \(\mathrm{N}\) on her and her parachute. (a) What is the weight of the parachutist? \((\mathrm{b})\) Draw a free-body diagram for the parachutist (see Section \(4.6 ) .\) Use that diagram to calculate the net force on the parachutist. Is the net force upward or downward? (c) What is the acceleration (magnitude and direction) of the parachutist?

Step-by-Step Solution

Answer

(a) Weight = 539.0 N downward. (b) Net force = 81 N upward. (c) Acceleration = 1.47 m/s² upward.

Step 1: Calculate the Weight of the Parachutist

The weight of an object is the force exerted by gravity on it. Weight is calculated using the formula: \( W = m \cdot g \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately \( 9.8\, \text{m/s}^2 \) on Earth's surface). For the parachutist, \( m = 55.0\, \text{kg} \).\[ W = 55.0\, \text{kg} \times 9.8\, \text{m/s}^2 = 539.0\, \text{N} \]Thus, the weight of the parachutist is 539.0 N, directed downward.

Step 2: Draw a Free-Body Diagram

In a free-body diagram, we represent all the forces acting on the parachutist. For the parachutist: - Downward force: weight of 539.0 N. - Upward force: air resistance of 620 N. These forces determine the net force acting on her.

Step 3: Calculate the Net Force

The net force is the difference between the upward and downward forces. Since the air resistance is 620 N upward and the weight is 539.0 N downward, the net force is:\[ F_{\text{net}} = F_{\text{up}} - F_{\text{down}} = 620\, \text{N} - 539.0\, \text{N} = 81\, \text{N} \]The net force of 81 N is upward.

Step 4: Determine the Acceleration

Using Newton's second law, \( F = m \cdot a \), solve for acceleration \( a \):\[ a = \frac{F_{\text{net}}}{m} = \frac{81\, \text{N}}{55.0\, \text{kg}} \approx 1.47\, \text{m/s}^2 \]Thus, the magnitude of the acceleration is \( 1.47\, \text{m/s}^2 \), directed upward.

Key Concepts

Net Force Calculation

Net force calculation is a fundamental concept in physics and it involves determining the total force acting on an object. In our scenario with the parachutist, we have two forces to consider:

• The weight of the parachutist, calculated as 539.0 N downward.
• The air resistance force, which is 620 N upward.

To find the net force, you subtract the smaller force from the larger force. This is because the forces are acting in opposite directions. The formula used is:\[F_{\text{net}} = F_{\text{up}} - F_{\text{down}}\]Substituting in our values gives:\[F_{\text{net}} = 620\, \text{N} - 539.0\, \text{N} = 81\, \text{N}\]This means the net force on the parachutist is 81 N upwards. Knowing the net force helps us understand how the movement and speed of the parachutist will change.

Newton's Second Law

Newton's second law is crucial for understanding how forces affect motion. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. The formula is:\[F = m \cdot a\]In our case, we need to rearrange this formula to solve for acceleration \(a\), giving us:\[a = \frac{F}{m}\]Newton's second law allows us to link force, mass, and acceleration, forming the backbone of how we calculate how quickly an object will speed up or slow down. This law is why the parachutist accelerates upward when the net upward force is applied.

Acceleration Determination

After finding the net force, the next step is to determine the acceleration of the parachutist. Using Newton's second law, we can calculate acceleration by dividing the net force by the mass.For the parachutist, we already established:

• Net force \(F_{\text{net}} = 81\, \text{N}\)
• Mass \(m = 55.0\, \text{kg}\)

Using the formula:\[a = \frac{F_{\text{net}}}{m} = \frac{81\, \text{N}}{55.0\, \text{kg}} \approx 1.47\, \text{m/s}^2\]This calculation shows that the parachutist accelerates at approximately 1.47 meters per second squared upward. Acceleration refers to how quickly the speed of the parachutist is changing, and since the net force is upwards, the acceleration is also in the same direction.