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Magazine Chapter 7: Problem 5
A father racing his son has half the kinetic energy of the son, who has half the mass of the father. The father speeds up by \(1.0 \mathrm{~m} / \mathrm{s}\) and then has the same kinetic energy as the son. What are the original speeds of (a) the father and (b) the son?
Short Answer
Expert verified
Father's original speed is 2 m/s, son's is 2.83 m/s.
Step by step solution
01
Understand the Given Information
The father's kinetic energy initially is half that of the son's kinetic energy, and the father's mass is double the son's mass. When the father speeds up by 1.0 m/s, his kinetic energy equals the son's kinetic energy.
02
Set Up Initial Equations
Let the mass of the son be \(m\) and the velocity of the son be \(v_s\). Then the mass of the father is \(2m\), and his initial velocity is \(v_f\). The kinetic energy of the son is \(\frac{1}{2}m v_s^2\), and the initial kinetic energy of the father is \(\frac{1}{2} \times \frac{1}{2}m v_s^2 = \frac{1}{4}m v_s^2\).
03
Develop the Father's Kinetic Energy Equation
Since the father has half the kinetic energy of the son, we can establish that:\[ \frac{1}{2}(2m)v_f^2 = \frac{1}{4}mv_s^2 \]This simplifies to:\[ 2m v_f^2 = mv_s^2 \]Therefore:\[ v_f^2 = \frac{1}{2}v_s^2 \]
04
Calculate Change in Velocity for the Father
The father speeds up by 1.0 m/s, so his new velocity is \((v_f + 1)\). The new kinetic energy is equal to the son's kinetic energy:\[ \frac{1}{2} (2m)(v_f + 1)^2 = \frac{1}{2}m v_s^2 \]Simplifying, we have:\[ (v_f + 1)^2 = v_s^2 \]
05
Solve for Velocities
From equation 3, \(v_f^2 = \frac{1}{2}v_s^2\). Since \((v_f + 1)^2 = v_s^2\), substituting \(v_f = \sqrt{\frac{1}{2}}v_s\), we solve:\[ (\sqrt{\frac{1}{2}}v_s + 1)^2 = v_s^2 \]This simplifies to:\[ \frac{1}{2}v_s^2 + 2\sqrt{\frac{1}{2}}v_s + 1 = v_s^2 \]Subtracting \(\frac{1}{2}v_s^2\) from both sides:\[ 2\sqrt{\frac{1}{2}}v_s + 1 = \frac{1}{2}v_s^2 \]
06
Solve Quadratic Equation
This simplies to a quadratic in terms of \(v_s\):\[ v_s^2 - 4\sqrt{\frac{1}{2}}v_s - 2 = 0 \]Solve quadratic using the quadratic formula \(v_s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\sqrt{\frac{1}{2}}\), and \(c = -2\).
07
Explain the Quadratic Formula Solution
Calculate \(b^2 - 4ac = 4\times(\frac{1}{2})\times4 \times 1 + 8 = 16 \), thus solutions will be real. Therefore:\[ v_s = \frac{4\sqrt{\frac{1}{2}} \pm \sqrt{16}}{2} \]Simplifying gives \(v_s\) as either 0 or positive \(4\sqrt{\frac{1}{2}} \approx 2.828\). However, as velocity cannot be zero, the son moves with a speed \(v_s = -b\approx 2.83 \text{ m/s}\).
08
Final Calculation and Verification
With son鈥檚 speed \(v_s=2.83 \text{ m/s}\), calculate father's speed using equation:\( v_f = \sqrt{\frac{1}{2}}v_s = 2 \). Increase father鈥檚 velocity by 1 m/s, verify father鈥檚 speed inward check: father = new son鈥檚 kinetic energy equals by plugging same kinetic energy as current grandson.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mass and Velocity
In physics, two crucial factors that influence an object's kinetic energy are its mass and velocity. Mass refers to the amount of matter in an object, typically measured in kilograms (kg). Velocity is a bit more dynamic; it represents the speed of an object in a particular direction and is measured in meters per second (m/s). Both mass and velocity are fundamental in determining the kinetic energy of an object.
In the context of our exercise, the father's mass is twice that of his son. This means that if the son's mass is denoted as \(m\), the father's mass would be \(2m\). Despite having a larger mass, the father's kinetic energy is initially less than the son's due to differences in their velocity.
Velocity changes can have a significant impact on the kinetic energy, as seen when the father increases his velocity by 1.0 m/s to match his son's kinetic energy. This illustrates how velocity adjustments play a key role in energy dynamics.
In the context of our exercise, the father's mass is twice that of his son. This means that if the son's mass is denoted as \(m\), the father's mass would be \(2m\). Despite having a larger mass, the father's kinetic energy is initially less than the son's due to differences in their velocity.
Velocity changes can have a significant impact on the kinetic energy, as seen when the father increases his velocity by 1.0 m/s to match his son's kinetic energy. This illustrates how velocity adjustments play a key role in energy dynamics.
Kinetic Energy Equations
Kinetic energy is a measure of the energy possessed by an object due to its motion. It is given by the equation \( KE = \frac{1}{2}mv^2 \), where \(m\) is the mass and \(v\) is the velocity. This formula indicates that kinetic energy is directly proportional to the mass of an object and the square of its velocity, highlighting the quadratic nature of velocity's influence on energy.
In our problem, the son's kinetic energy can be written as \( \frac{1}{2}mv_s^2 \). The father's initial kinetic energy is given by \( \frac{1}{4}mv_s^2 \), because it鈥檚 half that of his son. As the father increases his speed by 1.0 m/s, his new kinetic energy matches that of his son, showing how sensitive kinetic energy is to changes in velocity.
Understanding these equations helps us predict how kinetic energy will change with alterations in mass or velocity, which is fundamental in solving physics problems.
In our problem, the son's kinetic energy can be written as \( \frac{1}{2}mv_s^2 \). The father's initial kinetic energy is given by \( \frac{1}{4}mv_s^2 \), because it鈥檚 half that of his son. As the father increases his speed by 1.0 m/s, his new kinetic energy matches that of his son, showing how sensitive kinetic energy is to changes in velocity.
Understanding these equations helps us predict how kinetic energy will change with alterations in mass or velocity, which is fundamental in solving physics problems.
Quadratic Equation
When solving problems involving kinetic energy, especially those with changing velocities, quadratic equations often come into play. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \). It arises in physics when handling relationships where velocity is squared, as in the kinetic energy equation.
In this exercise, to find the velocity of the son, we end up with the quadratic equation \( v_s^2 - 4\sqrt{\frac{1}{2}}v_s - 2 = 0 \). To solve it, we use the quadratic formula: \[ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a\), \(b\), and \(c\) are constants from the equation. This approach allows us to find the velocity values that satisfy both the initial and changed conditions of kinetic energy.
Solving quadratic equations not only helps in understanding physics problems but also strengthens mathematical skills, which are valuable in various scientific contexts.
In this exercise, to find the velocity of the son, we end up with the quadratic equation \( v_s^2 - 4\sqrt{\frac{1}{2}}v_s - 2 = 0 \). To solve it, we use the quadratic formula: \[ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a\), \(b\), and \(c\) are constants from the equation. This approach allows us to find the velocity values that satisfy both the initial and changed conditions of kinetic energy.
Solving quadratic equations not only helps in understanding physics problems but also strengthens mathematical skills, which are valuable in various scientific contexts.
