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Chapter 2: Problem 36

What is the kinetic energy, in joules, of a \(625-\mathrm{kg}\) roller coaster car that is moving with a speed of \(18.3 \mathrm{~m} / \mathrm{s}\) ? A. \(1.05 \times 10^4 \mathrm{~J}\) B. \(1.05 \times 10^5 \mathrm{~J}\) C. \(2.05 \times 10^5 \mathrm{~J}\) D. \(1.06 \times 10^4 \mathrm{~J}\)

Short Answer

Expert verified
B. \(1.05 \times 10^5 \, \mathrm{J}\)

Step by step solution

01

Understand the Kinetic Energy Formula

The formula for kinetic energy is given by \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass of the object and \( v \) is its velocity.
02

Identify Given Values

From the problem, we know:\[ m = 625 \, \mathrm{kg} \]\[ v = 18.3 \, \mathrm{m/s} \]
03

Substitute the Values

Substitute the given values into the kinetic energy formula:\[ KE = \frac{1}{2} (625 \, \mathrm{kg}) (18.3 \, \mathrm{m/s})^2 \]
04

Calculate the Velocity Squared

First, calculate the square of the velocity:\[ v^2 = (18.3 \, \mathrm{m/s})^2 = 334.89 \, \mathrm{(m/s)^2} \]
05

Compute the Kinetic Energy

Now, multiply the mass by the squared velocity and then by \( \frac{1}{2} \):\[ KE = \frac{1}{2} (625 \, \mathrm{kg}) (334.89 \, \mathrm{(m/s)^2}) \]\[ KE = \frac{1}{2} (209306.25 \, \mathrm{kg \, (m/s)^2}) \]\[ KE = 104653.125 \, \mathrm{J} \]
06

Compare with Options

Compare the calculated kinetic energy to the options provided. The value closest to our result is:\[ 1.05 \times 10^5 \, \mathrm{J} \]

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Key Concepts

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

Physics Formulas
Understanding physics formulas is crucial for solving problems in physics. These formulas provide a structured way to calculate various physical properties. In this exercise, we use the kinetic energy formula, which is:
  • Kinetic Energy (KE) = \[ \frac{1}{2}mv^2 \].
  • Here, KE is the kinetic energy, m is the object's mass, and v is the velocity.
This formula helps us understand how mass and velocity affect an object's energy. By substituting the given values into the formula, we can solve for the kinetic energy easily.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. If an object is moving, it has kinetic energy, and the amount of this energy depends on two main factors:
  • Mass: The heavier the object, the more kinetic energy it has.
  • Velocity: The faster the object is moving, the more kinetic energy it has.
This type of energy is calculated using the formula \[ KE = \frac{1}{2}mv^2 \]. For instance, in our problem:m = 625 \mathrm{kg} and v = 18.3 \mathrm{m/s}. By computing these values, we find the kinetic energy of the roller coaster car. This helps us understand how energy changes with motion.
Motion Equations
In physics, motion equations describe the relationship between an object's movement, its speed, and the forces acting upon it. The kinetic energy formula is derived from these motion equations and principles. To solve our given problem, let's follow these steps:
  • Step 1: Square the velocity value. \(v^2 = (18.3 \mathrm{m/s})^2 = 334.89 \mathrm{(m/s)^2}\).
  • Step 2: Incorporate this squared velocity into the kinetic energy formula. \[ KE = \frac{1}{2} (625 \mathrm{kg})(334.89 \mathrm{(m/s)^2}) \].
  • Step 3: Calculate the kinetic energy. KE= \[ 104653.125 \mathrm{J} \] (joules).
These steps show how motion equations guide us to determine the kinetic energy of an object. Understanding each step improves comprehension of the broader concepts in physics.

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Most popular questions from this chapter

The power of a motor pump is \(2 \mathrm{~kW}\). How much water per minute can the pump raise to a height of \(10 \mathrm{~m}\) ? (Under the influence of gravitational force, work = mass \((\mathrm{m}) \times\) gravitational field strength \((g) \times\) height \((h)\), where \(g=10 \mathrm{~ms}^{-2}\).) A. \(1,200 \mathrm{~kg}\) B. \(1,200 \mathrm{~g}\) C. \(120 \mathrm{~kg}\) D. \(120 \mathrm{~g}\)

In warm, humid weather, people who wear glasses experience a very common effect of phase change while getting off an air-conditioned bus: their glasses suddenly fog up. After a while, the glasses become clear again. Which of the following physical processes are involved in these phenomena? A. condensation followed by evaporation B. condensation followed by fusion C. solidification followed by evaporation D. solidification followed by fusion

A scientist places \(10 \mathrm{~mL}\) of water in a test tube and heats it until the liquid boils and escapes as steam. This is an example of which of the following? A. chemical change involving changes of state B. chemical change involving chemical reactions C. physical change involving changes of state D. physical change involving chemical reactions

Which of the following describes an ionic bond? A. Two atoms of the same element share a pair of electrons in the middle of the two atoms. B. One metallic and one nonmetallic atom bond by attracting oppositely charged ions. C. Two metallic atoms bond through the electrostatic attractive force between conduction electrons and positively charged metal ions. D. Two nonmetallic atoms bond by sharing their electrons.

Hydroelectric dams make use of the differences in water levels to generate electricity. Which of the following shows the energy conversion in a dam? A. Gravitational potential energy \(\rightarrow\) Kinetic energy \(\rightarrow\) Electrical energy B. Electrical energy \(\rightarrow\) Heat energy \(\rightarrow\) Electrical energy
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