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 -g baseball at \(60.0 \mathrm{~m} / \mathrm{s}\) c) a \(20.0-\mathrm{g}\) bullet at \(300 . \mathrm{m} / \mathrm{s}\)

Short answer

Answer: The stone has the highest kinetic energy on impact, with a kinetic energy of 4500 J. The bullet has a kinetic energy of 900 J, and the baseball has a kinetic energy of 180 J.

Step-by-step solution

Calculating the kinetic energy of the 10.0-kg stone

First, let's find the kinetic energy of the stone. Given its mass (m) of 10.0 kg and velocity (v) of 30.0 m/s, we can use the following formula for kinetic energy: \(KE = \frac{1}{2}mv^2\). \(KE_{stone} = \frac{1}{2}(10.0\,\mathrm{kg})(30.0\,\mathrm{m/s})^2 = 4500\,\mathrm{J}\) Joules.

Calculating the kinetic energy of the 100.0-g baseball

Next, let's find the kinetic energy of the baseball. We need to convert its mass into kilograms first: 100.0 g = 0.1 kg. Given its mass (m) of 0.1 kg and velocity (v) of 60.0 m/s, we can use the following formula for kinetic energy: \(KE = \frac{1}{2}mv^2\). \(KE_{baseball} = \frac{1}{2}(0.1\,\mathrm{kg})(60.0\,\mathrm{m/s})^2 = 180\,\mathrm{J}\) Joules.

Calculating the kinetic energy of the 20.0-g bullet

Lastly, let's find the kinetic energy of the bullet. We need to convert its mass into kilograms first: 20.0 g = 0.02 kg. Given its mass (m) of 0.02 kg and velocity (v) of 300.0 m/s, we can use the following formula for kinetic energy: \(KE = \frac{1}{2}mv^2\). \(KE_{bullet} = \frac{1}{2}(0.02\,\mathrm{kg})(300.0\,\mathrm{m/s})^2 = 900\,\mathrm{J}\) Joules.

Comparing the three kinetic energies

Now, let's compare the kinetic energies of the three projectiles: - Stone: \(KE_{stone} = 4500\,\mathrm{J}\) - Baseball: \(KE_{baseball} = 180\,\mathrm{J}\) - Bullet: \(KE_{bullet} = 900\,\mathrm{J}\) The stone has the highest kinetic energy on impact, followed by the bullet, and then the baseball.

Key concepts

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. The motion of a projectile can be studied using two components: horizontal and vertical motions.

The horizontal motion is constant, as there is typically no acceleration in the horizontal direction (ignoring air resistance). The vertical motion, on the other hand, is affected by the acceleration due to gravity, which causes the projectile to follow a parabolic path. In calculating the impact of a projectile, the velocity at the exact moment of impact is crucial, as this determines its kinetic energy.

Most problems on projectile motion are solved by breaking down the initial velocity into horizontal and vertical components, using the kinematic equations to find the range, maximum height, and time of flight. However, for kinetic energy, we focus on the speed at impact, which combines both components of motion.

Kinetic Energy Formula

The kinetic energy (KE) of an object is the energy it possesses due to its motion. The formula to calculate kinetic energy is: \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass of the object and \( v \) is the speed or velocity of the object.

The unit for measuring kinetic energy is the joule (J) in the International System of Units (SI). When solving problems that involve kinetic energy, it is important to ensure that the units of mass are in kilograms (kg) and speed is in meters per second (m/s), which are the standard SI units.

The value of kinetic energy increases quadratically with the velocity; doubling the speed of an object will increase its kinetic energy by a factor of four, emphasizing the profound impact of velocity on kinetic energy.

Energy Comparison

When comparing the kinetic energies of different objects, several factors need to be taken into account, primarily mass and velocity. Since kinetic energy is directly proportional to the mass of an object and to the square of its velocity, even a small object can have a large kinetic energy if its velocity is high.

Let's analyze the projectiles we calculated earlier - a heavier stone has much more kinetic energy than the lighter baseball and bullet, though it is moving at a slower speed than the bullet. This illustrates how the mass of an object can significantly influence its kinetic energy. Conversely, a bullet, despite having a smaller mass, achieves considerable kinetic energy due to its extremely high velocity, demonstrating the velocity's squared relationship with kinetic energy.

Comparing the kinetic energy of objects helps us to understand the potential impact they can have upon collision. In practical applications, this allows us to predict the damage potential of projectiles or the effectiveness of different masses moving at various speeds, which is essential for understanding safety protocols, designing effective safety features, and analyzing events in the physical world.