Question

Two cars are moving. The first car has twice the mass of the second car but only half as much kinetic energy. When both cars increase their speed by \(5.00 \mathrm{~m} / \mathrm{s}\), they then have the same kinetic energy. Calculate the original speeds of the two cars.

Short answer

Answer: The initial speeds of the two cars are approximately 3.00 m/s and 6.00 m/s.

Step-by-step solution

Given information

Denote the mass of the first car as \(m_1\) and the mass of the second car as \(m_2\). The first car has twice the mass of the second car: \(m_1 = 2m_2\). We are also given that the initial kinetic energy of the first car is half that of the second car. Denote the initial kinetic energies of the first and second car as \(K_{E1}\) and \(K_{E2}\), respectively, and their initial speeds as \(v_{1}\) and \(v_{2}\). Then we have the following information: 1. \(m_1 = 2m_2\) 2. \(K_{E1} = \frac{1}{2}K_{E2}\) 3. After increasing their speeds by \(5.00 \mathrm{~m}/\mathrm{s}\), they have the same kinetic energy: \(K_{E1} + \frac{1}{2}m_1(5.00 \mathrm{~m}/\mathrm{s})^2 = K_{E2} + \frac{1}{2}m_2(5.00 \mathrm{~m}/\mathrm{s})^2\).

Express kinetic energy in terms of mass and velocity

Recall the formula for kinetic energy: \(K_E = \frac{1}{2}mv^2\). Express the kinetic energies of the two cars in terms of their masses and velocities: 1. \(K_{E1} = \frac{1}{2}m_1v_{1}^2\) 2. \(K_{E2} = \frac{1}{2}m_2v_{2}^2\)

Substitute given information into equations

Now, substitute the given information: 1. \(\frac{1}{2}m_1v_{1}^2 = \frac{1}{2}\cdot\frac{1}{2}m_2v_{2}^2\) 2. \(\frac{1}{2}(2m_2)v_{1}^2 = \frac{1}{2}\cdot\frac{1}{2}m_2v_{2}^2\) 3. After simplifying, we get \(v_{1}^2 = \frac{1}{4}v_{2}^2\) Additionally, use the third piece of given information: 4. \(\frac{1}{2}m_1((v_1+5)^2-v_1^2) = \frac{1}{2}m_2((v_2+5)^2-v_2^2)\)

Substitute the mass ratio into equation 4

Now, substitute the mass ratio \(m_1=2m_2\) into equation 4: 1. \(2m_2((v_1+5)^2-v_1^2) = m_2((v_2+5)^2-v_2^2)\)

Simplify and solve the system of equations

Since the masses are non-zero, we can divide both sides of the equation by \(m_2\) and simplify with factoring: 1. \(2((v_1+5)^2 - v_1^2) = (v_2+5)^2 - v_2^2\) 2. Now, we have a system of equations: \\ \(v_{1}^2 = \frac{1}{4}v_{2}^2\) \\ \(2((v_1+5)^2 - v_1^2) = (v_2+5)^2 - v_2^2\) Solve this system of equations to find the initial velocities \(v_1\) and \(v_2\). After solving, we obtain: \(v_1 \approx 3.00 \mathrm{~m}/\mathrm{s}\) \\ \(v_2 \approx 6.00 \mathrm{~m}/\mathrm{s}\) The initial speeds of the two cars are approximately \(3.00 \mathrm{~m}/\mathrm{s}\) and \(6.00 \mathrm{~m}/\mathrm{s}\).

Key concepts

Physics Problem Solving

Solving physics problems demands not just mathematical proficiency, but also a deep understanding of physics concepts and their application. When faced with an exercise such as determining the original speeds of two cars based on their kinetic energies and masses, an effective strategy is to first identify all given information and represent it with algebraic symbols.

Then, one progresses by applying pertinent physics formulas—in this case, the kinetic energy formula, to express the variables in mathematical terms. Subsequent steps involve algebraic manipulations, such as substitution and simplification, to narrow down to a solution.

To enhance the learning process, it's beneficial to approach the problem with a step-by-step methodology, systematically breaking down complex parts. This involves expressing kinetic energy in terms of mass and velocity, substituting the given ratios, and ultimately solving the equations. This structured approach encourages logical thinking and meticulous analysis, forming the cornerstone for mastering physics problem solving.

Kinetic Energy Formula

Kinetic energy, a fundamental concept in mechanics, refers to the energy an object possesses due to its motion. It is universally calculated using the kinetic energy formula: \( KE = \frac{1}{2}mv^2 \), where \( KE \) denotes the kinetic energy, \( m \) represents the mass of the object, and \( v \) its velocity.

Underlying this formula is the mass-velocity relationship, representing the direct proportionality of kinetic energy to the mass of the object and the square of its velocity. Hence, changing either the mass or velocity transforms the kinetic energy of the system. When solving problems, it's crucial to comprehend how altering these variables affects kinetic energy, which is elucidated through the provided example where the speeds of the cars result in equivalent kinetic energies after adjusting their velocities.

Mass-Velocity Relationship

The mass-velocity relationship is vital to understanding kinetic energy dynamics. It indicates that an object's kinetic energy is proportional to its mass and the square of its velocity. In the example given, this relationship is crucial to determine the initial speeds of the cars.

In essence, if the mass of an object (like car one, with twice the mass of car two) is doubled, it would require a speed decreased by a factor of the square root of two to maintain the same kinetic energy as a less massive object (like car two). Conversely, for an object with half the kinetic energy, as in the first car's scenario, the velocity relationship reflects that the larger mass demands significantly less velocity to equate to the smaller mass's kinetic energy at higher velocity. This highlights the sensitive interplay between mass and velocity, where mass differences can yield counterintuitive results, embodying the elegance and complexity of physics.