Question

Josh, who has a mass of \(80 \mathrm{kg}\), and Sarah, who has a mass of \(50 \mathrm{kg}\) jump off a \(20 \mathrm{m}\) tall building and land on a fire net. The net compresses, and they bounce back up at the same time. Which of the following statements is NOT true? (A) Sarah will bounce higher than Josh. (B) For Josh, the change in speed from the start of the jump to contacting the net is \(30 \frac{\mathrm{m}}{\mathrm{s}}\) (C) Josh will experience a greater force upon impact than Sarah. (D) The energy in this event is converted from potential to kinetic to elastic to kinetic.

Short answer

(B) For Josh, the change in speed from the start of the jump to contacting the net is 30 m/s

Step-by-step solution

Title - Calculate the speed just before contact

To find the speed just before contacting the net, use the formula for the free-fall velocity: \[ v = \sqrt{2gh} \] where: - v is the velocity- g is the acceleration due to gravity (\(9.8 \frac{m}{s^{2}}\))- h is the height (20 m) Substituting the values: \[ v = \sqrt{2 * 9.8 \frac{m}{s^{2}} * 20 \mathrm{m}} \]\[ v = \sqrt{392} \approx 19.8 \frac{m}{s} \]Therefore, the speed just before contacting the net is approximately 19.8 m/s.

Content - Analyze the given statements

Now evaluate each statement: (A) Sarah will bounce higher than Josh.Since both Josh and Sarah land on the same net and assuming it behaves identically for both, this statement could be true if not considering their masses and energy conversion efficiency.(B) For Josh, the change in speed from the start of the jump to contacting the net is 30 m/s.We calculated the speed to be about 19.8 m/s, much less than 30 m/s. Therefore, this statement is NOT true.(C) Josh will experience a greater force upon impact than Sarah.This statement holds true since force equals mass times acceleration, and Josh has a greater mass than Sarah.(D) The energy in this event is converted from potential to kinetic to elastic to kinetic.This is true as it describes the transformation of energy from jumping off to bouncing back.

Conclusion - Identify the incorrect statement

The results show that Statement (B) is incorrect. Josh's speed just before contacting the net was 19.8 m/s and not 30 m/s as claimed. Therefore, the statement (B) is false.

Key concepts

Free Fall Velocity Calculation

In this exercise, we need to calculate the velocity that Josh and Sarah reach just before they contact the fire net after jumping from a building. This is known as the 'free fall velocity.'
To find this, we use the formula for free fall velocity, which is given by:
\[ v = \sqrt{2gh} \]
where:

• \( v \) is the velocity
• \( g \) is the acceleration due to gravity, approximately \(9.8 \frac{m}{s^{2}}\)
• \( h \) is the height from which they jump—in this case, 20 meters

By substituting the height and the gravitational constant into the formula:
\[ v = \sqrt{2 \times 9.8 \frac{m}{s^{2}} \times 20 m} \]
we get:
\[ v = \sqrt{392} \approx 19.8 \frac{m}{s} \]
This means that just before they hit the net, both Josh and Sarah are falling at a speed of approximately 19.8 meters per second.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. When Josh and Sarah are falling, their potential energy is converted into kinetic energy.
The kinetic energy (KE) of an object can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

For Josh, who has a mass of 80 kg, the kinetic energy just before hitting the net would be:
\[ KE_{Josh} = \frac{1}{2} \times 80 kg \times (19.8 \frac{m}{s})^2 \]
We calculate this as:
\[ KE_{Josh} = 0.5 \times 80 \times 392.04 \approx 15681.6 J \]
Similarly, for Sarah, who has a mass of 50 kg:
\[ KE_{Sarah} = \frac{1}{2} \times 50 kg \times (19.8 \frac{m}{s})^2 \]
This comes out to:
\[ KE_{Sarah} = 0.5 \times 50 \times 392.04 \approx 9801 J \]
Thus, Josh has more kinetic energy than Sarah due to his greater mass.

Potential Energy

Potential energy is the energy stored in an object due to its position or height. When Josh and Sarah are on top of the 20-meter-high building, they have potential energy due to gravity.
The formula for gravitational potential energy (PE) is:
\[ PE = mgh \]
where:

• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity, approximately \(9.8 \frac{m}{s^{2}}\)
• \( h \) is the height

For Josh, the potential energy at the top of the building is:
\[ PE_{Josh} = 80 kg \times 9.8 \frac{m}{s^{2}} \times 20 m \]
Let's calculate this:
\[ PE_{Josh} = 80 \times 9.8 \times 20 \approx 15680 J \]
For Sarah, the potential energy is:
\[ PE_{Sarah} = 50 kg \times 9.8 \frac{m}{s^{2}} \times 20 m \]
This comes out to:
\[ PE_{Sarah} = 50 \times 9.8 \times 20 \approx 9800 J \]
Both Josh and Sarah possess potential energy before they jump, which subsequently transforms into kinetic energy during the fall.

Force and Mass Relationship

In this scenario, the force experienced by Josh and Sarah upon impacting the fire net can be analyzed using Newton's second law, which states:
\[ F = ma \]
where:

• \( F \) is the force
• \( m \) is the mass of the object
• \( a \) is the acceleration

Since Josh has a greater mass than Sarah, he will experience a greater force upon impact if they both decelerate at the same rate when they hit the net.
To break it down, let's suppose they both experience the same deceleration from the net. The force on Josh would be:
\[ F_{Josh} = 80 kg \times a \]
and the force on Sarah would be:
\[ F_{Sarah} = 50 kg \times a \]
Clearly, since 80 kg is greater than 50 kg, the force \( F_{Josh} \) is larger than \( F_{Sarah} \). Therefore, Josh experiences a greater force due to his higher mass.
Hence, this explains why the statement that 'Josh will experience a greater force upon impact than Sarah' is indeed true.