Question

The damage done by a projectile on impact is correlated with its kinetic energy. Calculate and compare the kinetic energies of these three projectiles: a) a \(10.0 \mathrm{~kg}\) stone at \(30.0 \mathrm{~m} / \mathrm{s}\) b) a \(100.0 \mathrm{~g}\) baseball at \(60.0 \mathrm{~m} / \mathrm{s}\) c) a \(20.0 \mathrm{~g}\) bullet at \(300 . \mathrm{m} / \mathrm{s}\)

Short answer

Question: Compare the kinetic energies of the three projectiles: a 10.0 kg stone at 30.0 m/s, a 100.0 g baseball at 60.0 m/s, and a 20.0 g bullet at 300.0 m/s. Answer: The stone has the highest kinetic energy (4500 J), followed by the bullet (900 J), and lastly the baseball (180 J).

Step-by-step solution

Understand the exercise and given information

We are supposed to calculate the kinetic energies of three projectiles and compare them. We have given masses and velocities for all three projectiles: a) 10.0 kg stone at 30.0 m/s b) 100.0 g baseball at 60.0 m/s c) 20.0 g bullet at 300.0 m/s

Convert masses to appropriate units

Since the base unit of mass in the SI system is kilograms, we should convert the grams (baseball and bullet) to kg before calculating the kinetic energy: b) 100.0 g to kg: \(\frac{100.0}{1000}=0.1\,kg\) c) 20.0 g to kg: \(\frac{20.0}{1000}=0.02\,kg\)

Calculate kinetic energies

Now we can calculate the kinetic energies (\(KE = \frac{1}{2}mv^2\)) for each projectile: a) Stone: \(KE_{stone} = \frac{1}{2}(10.0\,kg)(30.0\,m/s)^2 = 4500 \,J\) b) Baseball: \(KE_{baseball} = \frac{1}{2}(0.1\,kg)(60.0\,m/s)^2 = 180\,J\) c) Bullet: \(KE_{bullet} = \frac{1}{2}(0.02\,kg)(300.0\,m/s)^2 = 900\,J\)

Compare the kinetic energies

We have found the kinetic energies for all projectiles, now let's compare their values: - Stone: 4500 J - Baseball: 180 J - Bullet: 900 J In comparison, the stone has the highest kinetic energy, followed by the bullet, and lastly the baseball.

Key concepts

Projectile Motion

Projectile motion involves the movement of objects that are launched into the air and are affected by gravity and other forces. When a projectile is launched, it follows a curved path called a trajectory. The path is determined by the initial velocity of the projectile, the angle of launch, and the forces of gravity and air resistance. In the context of kinetic energy, which measures the energy an object possesses due to its motion, projectiles with different masses and velocities will have varying kinetic energies. This is crucial because the kinetic energy impacts how much damage can be done on impact.
Understanding projectile motion helps us calculate important aspects such as the time of flight, maximum height reached, and the distance traveled. Each of these parameters depends on the properties of the initial launch and the physical characteristics of the projectile itself, such as its weight and speed.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. In physics, it's essential to understand how kinetic energy—the energy an object possesses due to motion—can be converted from other energy forms, like potential energy. For instance, a stone held at a certain height has gravitational potential energy. When it is dropped, this potential energy converts into kinetic energy as it falls.
For projectiles, when they impact another surface or come to rest, their kinetic energy can be transferred into other forms, such as sound, heat, or deformation energy (causing a dent or breakage). This conversion is critical in understanding the total energy changes in the system and predicting the outcome of collisions. The conservation of energy principle states that energy cannot be created or destroyed; it only changes form. Thus, analyzing the energy conversion processes helps us track how energy is distributed throughout all parts of a system.

Physics Calculations

Physics calculations are fundamental in solving problems related to motion, energy, and forces. In the context of measuring kinetic energy, it's vital to correctly apply formulas and convert units for accurate results. The kinetic energy of an object can be calculated using the equation: \[ KE = \frac{1}{2}mv^2 \]where \(m\) is the mass of the object in kilograms and \(v\) is the velocity in meters per second.
For our projectiles, it was necessary to ensure the masses were in the appropriate unit (kilograms) before applying the formula. By calculating kinetic energy, we gain insights into how different masses and speeds affect the energy output in physical interactions.
It's also crucial to compare energies to understand the potential impact magnitude of different projectiles. Physics calculations not only allow us to quantify kinetic energy but also provide a systematic approach to solving complex problems methodically.