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Catapulting mushrooms. Certain mushrooms launch their spores by a catapult mechanism. As water condenses from the air onto a spore that is attached to the mushroom, a drop grows on one side of the spore and a film grows on the other side. The spore is bent over by the drop's weight, but when the film reaches the drop, the drop's water suddenly spreads into the film and the spore springs upward so rapidly that it is slung off into the air. Typically, the spore reaches a speed of \(1.6\) \(\mathrm{m} / \mathrm{s}\) in a \(5.0 \mu \mathrm{m}\) launch; its speed is then reduced to zero in \(1.0 \mathrm{~mm}\) by the air. Using those data and assuming constant accelerations, find the acceleration in terms of \(g\) during (a) the launch and (b) the speed reduction.

Step by step solution

01

Identify given information for the launch phase

We are given the speed of the spore after launch, \(v = 1.6 \, \text{m/s}\), and the distance over which this speed is achieved, \(s = 5.0 \, \mu\text{m} = 5.0 \times 10^{-6} \, \text{m}\). The initial speed \(u = 0 \, \text{m/s}\) because the spore starts from rest.

02

Calculate acceleration during the launch phase

Using the equation \(v^2 = u^2 + 2as\), where \(u = 0\), solving for \(a\) gives us:\[ a = \frac{v^2 - u^2}{2s} = \frac{(1.6)^2 - 0}{2 \times 5.0 \times 10^{-6}} \] Calculating this, we find:\[ a = \frac{2.56}{1.0 \times 10^{-5}} = 2.56 \times 10^5 \, \text{m/s}^2 \]

03

Express acceleration during launch in terms of g

We know that \(g = 9.81 \, \text{m/s}^2\). To express the launch acceleration \(a\) in terms of \(g\), we divide it by \(g\):\[ a_g = \frac{2.56 \times 10^5}{9.81} \approx 26,104.99 \, g \]

04

Identify given information for the speed reduction phase

For the speed reduction phase, the initial speed \(u = 1.6 \, \text{m/s}\) and the final speed \(v = 0 \, \text{m/s}\). The spore comes to rest over a distance \(s = 1.0 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m}\).

05

Calculate acceleration during the speed reduction phase

Using the same equation, \(v^2 = u^2 + 2as\), solve for \(a\):\[ 0 = (1.6)^2 + 2a(1.0 \times 10^{-3}) \]Rearranging gives:\[ a = -\frac{2.56}{2 \times 1.0 \times 10^{-3}} = -\frac{2.56}{2 \times 10^{-3}} = -1280 \, \text{m/s}^2 \]

06

Express acceleration during speed reduction in terms of g

Again, convert \(a\) to terms of \(g\) by dividing by \(9.81\):\[ a_g = \frac{-1280}{9.81} \approx -130.47 \, g \]

07

Conclusion for both phases

For the launch phase, the acceleration is approximately \(26,105 \, g\) upward. For the speed reduction phase, the acceleration is approximately \(-130.47 \, g\) downward. The negative sign indicates deceleration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acceleration

Understanding acceleration is crucial in analyzing the behavior of moving objects, including spores ejected from mushrooms. Acceleration describes how quickly an object changes its velocity. In this context, mushrooms utilize a powerful catapult mechanism to launch their spores with an impressive acceleration rate. As water gathers on the spore, tension builds until it's released, propelling the spore forward. Here are a few key points to remember about acceleration:

• It is a vector quantity, meaning it has both magnitude and direction.
• An object's acceleration can change as it moves, influenced by forces acting upon it.
• In physics, we calculate acceleration using the equation: \( a = \frac{v^2 - u^2}{2s} \), where \(v\) is final velocity, \(u\) is initial velocity, and \(s\) is distance traveled.

Through such calculations, the spore's acceleration during the launch phase is found to be significantly high, about 26,105 times the gravitational acceleration \(g\). This powerful acceleration enables spores to escape from the mushroom and disperse effectively.

Projectile Motion

Projectile motion involves objects moving under the influence of gravity, following a curved path. When a spore is catapulted from a mushroom, it experiences projectile motion. Several factors affect this type of motion:

• Initial Velocity: The initial push or speed given to the projectile.
• Angle of Launch: The angle at which the projectile is released determines its trajectory.
• Air Resistance: This can slow down the projectile, reducing the range it covers.

Once in motion, the spore follows a parabolic trajectory due to the influence of gravity. During this time, it moves horizontally and vertically, with its vertical motion being uniformly accelerated by gravity. This interaction makes projectile motion quite fascinating, especially for educational exercises involving spores, where understanding motion helps us comprehend natural dispersal mechanisms better.

Deceleration

Deceleration is an important aspect of motion, referring to a reduction in speed or velocity over a period of time. In the case of our spore, after being launched, it eventually slows down because of the resistance offered by the air around it. Here are some important aspects of deceleration:

• It is considered negative acceleration since the change in velocity is in the opposite direction of motion.
• The formula for calculating deceleration remains the same as acceleration: \( a = \frac{v^2 - u^2}{2s} \). A negative result indicates a slowing down or deceleration.
• Air resistance plays a critical role in bringing the spore to rest over a defined distance of 1.0 mm in this scenario, leading to a deceleration of around -130.47 \(g\).

Understanding deceleration helps us gain insights into how spores are able to come to a stop, demonstrating the delicate balance between natural forces and the engineered effectiveness of biological systems.