Question

An object of mass \(1000 \mathrm{~kg}\), initially having a velocity of \(100 \mathrm{~m} / \mathrm{s}\), decelerates to a final velocity of \(20 \mathrm{~m} / \mathrm{s}\). What is the change in kinetic energy of the object, in \(\mathrm{kJ}\) ?

Short answer

The change in kinetic energy is 300 kJ.

Step-by-step solution

Understand Kinetic Energy Formula

The kinetic energy (KE) of an object is given by the formula: \[ KE = \frac{1}{2}mv^{2} \]where 'm' is the mass of the object and 'v' is its velocity.

Calculate Initial Kinetic Energy

Using the initial velocity (\(v_i = 100 \text{ m/s}\)) and mass (\(m = 1000 \text{ kg}\)), calculate the initial kinetic energy: \[ KE_i = \frac{1}{2} (1000 \text{ kg}) (100 \text{ m/s})^2 = 500 \times 10000 = 500000 \text{ J} \]

Calculate Final Kinetic Energy

Using the final velocity (\(v_f = 20 \text{ m/s}\)) and mass (\(m = 1000 \text{ kg}\)), calculate the final kinetic energy: \[ KE_f = \frac{1}{2} (1000 \text{ kg}) (20 \text{ m/s})^2 = 500 \times 400 = 200000 \text{ J} \]

Determine Change in Kinetic Energy

Subtract the final kinetic energy from the initial kinetic energy: \[ \Delta KE = KE_i - KE_f = 500000 \text{ J} - 200000 \text{ J} = 300000 \text{ J} \]

Convert Joules to Kilojoules

To convert the change in kinetic energy from Joules to Kilojoules, divide by 1000: \[ \Delta KE = \frac{300000 \text{ J}}{1000} = 300 \text{ kJ} \]

Key concepts

kinetic energy formula

Kinetic energy (KE) measures the energy that an object possesses thanks to its motion. The formula is straightforward: i.e., KE = \( \frac{1}{2} mv^2 \) where:

• 'm'
  mass of the object.
• 'v'
  velocity of the object.

Let's break this down: Mass is how much stuff the object contains. Velocity is how fast the object is moving and in what direction. The square of velocity tells us that a small change in speed can cause a larger change in kinetic energy.As such, an object’s kinetic energy depends heavily on its speed. If either mass or velocity of the object changes, its kinetic energy changes.

initial velocity

Initial velocity (\(v_i\)) is the speed at which an object starts moving before any force affects it. In problems involving change in kinetic energy, knowing the initial velocity is essential because it helps calculate the initial amount of kinetic energy. In our example, the object initially travels at \(v_i = 100 m/s\). Plugging this into the kinetic energy formula with the mass \((m = 1000 kg)\) provides the initial KE: \( KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} \times 1000 \times (100)^2 = 500000 J \). Knowing the initial kinetic energy helps to determine how much energy the object has lost (or gained) during its motion.

final velocity

Final velocity (\(v_f\)) represents the speed of the object after it has been influenced by forces such as friction or applied force. It helps in finding the new kinetic energy after an event has occurred. In our problem, the object’s final velocity is \(v_f = 20 m/s\). The final kinetic energy is calculated as: \( KE_f = \frac{1}{2} m v_f^2 = \frac{1}{2} \times 1000 \times (20)^2 = 200000 J \). This lower kinetic energy tells us that the object has slowed down significantly, which is specified in \(J\) or Joules.

mass

Mass (\(m\)) is a critical part of the kinetic energy equation. It refers to how much matter makes up the object. Whether we're talking about planets, cars, or tiny particles, mass is always crucial. In our problem, the mass remains constant at \(m = 1000 kg\).The mass affects the total kinetic energy because: \( KE \text{ is directly proportional to } m \)(i.e. doubling the mass keeps KE doubled, assuming velocity is constant). Therefore, accurately knowing an object's mass is necessary for correctly calculating kinetic energy.

energy conversion

Energy conversion describes the process of changing one form of energy into another. In our problem, kinetic energy is converted into thermal energy, potential energy, or both due to the object slowing down, caused by friction or other forces. The difference between the initial and final kinetic energy represents this conversion.Thus, when the kinetic energy decreases from \(500000 J\) to \(200000 J\), it means \(300000\) Joules of energy have converted to other energy forms. We express this change in kilojoules for convenience: \(1 kJ = 1000 J\). Hence, \(300000 J\) = \(300 kJ\). Understanding this helps link kinetic energy dynamics to real-world phenomena.