Problem 10 - University Physics with Modern Physics

Question

A stone of mass \(m\) is thrown upward at an angle \(\theta\) above the horizontal and feels no appreciable air resistance. Use conservation of energy to show that at its highest point, it is a distance \(v_{0}^{2}\left(\sin ^{2} \theta\right) / 2 g\) above the point where it was launched. (Hint: \(v_{0}^{2}=v_{\mathrm{ax}}^{2}+v_{\mathrm{oy}}^{2} . )\)

Step-by-Step Solution

Answer

The stone reaches a height of \( \frac{v_0^2 \sin^2 \theta}{2g} \) at its highest point.

Step 1: Identify Initial Energies

To use the conservation of energy, identify the initial kinetic energy (KE) of the stone. The initial speed is given by \( v_0 \), and the kinetic energy is calculated as \( KE_i = \frac{1}{2}mv_0^2 \).The initial potential energy (PE) is zero because the stone starts from the launch point, which we assume to be the reference level where potential energy is zero. Thus, \( PE_i = 0 \).

Step 2: Break Down Initial Velocity into Components

The initial velocity \( v_0 \) can be broken into horizontal and vertical components using the angle \( \theta \).- Horizontal component: \( v_{0x} = v_0 \cos \theta \)- Vertical component: \( v_{0y} = v_0 \sin \theta \)

Step 3: Determine Mechanical Energy at the Highest Point

At the highest point, the vertical component of the velocity is zero \( (v_{y} = 0) \), but the horizontal component remains, so:- Horizontal velocity at the top: \( v_{x} = v_0 \cos \theta \)Using conservation of mechanical energy, the mechanical energy at the top consists of the horizontal kinetic energy alone, since the vertical kinetic energy is zero.Thus, \( KE_t = \frac{1}{2}mv_{x}^2 = \frac{1}{2}m(v_0 \cos \theta)^2. \)The potential energy at the top is what we need to determine.

Step 4: Apply Conservation of Energy

According to conservation of energy:Initial total energy = Total energy at the highest point.\[ KE_i + PE_i = KE_t + PE_t. \]Substituting the known values,\[ \frac{1}{2}mv_0^2 = \frac{1}{2}m(v_0 \cos \theta)^2 + mgh. \]Solve for \( h \), the height:\[ mv_0^2 - m(v_0 \cos \theta)^2 = 2mgh. \]

Step 5: Solve for Height

Simplify the equation from Step 4:\[ mv_0^2 - mv_0^2 \cos^2 \theta = 2mgh \]\[ mv_0^2 (1 - \cos^2 \theta) = 2mgh \]\(1 - \cos^2 \theta = \sin^2 \theta \), so:\[ mv_0^2 \sin^2 \theta = 2mgh \].Solve for \( h \):\[ h = \frac{v_0^2 \sin^2 \theta}{2g}. \]

Step 6: Conclusion

We have shown that at its highest point, the stone reaches a height of \( \frac{v_0^2 \sin^2 \theta}{2g} \) above the launch point, as required by the problem statement.

Key Concepts

Projectile Motion

Projectile motion is a fascinating concept in physics. It describes the motion of an object that is thrown or projected into the air, subject to only the acceleration of gravity. Such motion occurs in a curved path, and this path is known as a trajectory. For a projectile launched at an angle \( \theta \) to the horizontal, its initial velocity \( v_0 \) can be split into two components:

• The horizontal component \( v_{0x} = v_0 \cos \theta\) remains constant throughout the motion, as there is no horizontal acceleration.
• The vertical component \(v_{0y} = v_0 \sin \theta\) decreases upward until the highest point and then increases in the downward direction due to gravity.

As the stone reaches its highest point in this problem, the vertical velocity is zero, which allows us to find the maximum height using energy principles.

Kinetic Energy

Kinetic energy (KE) is the energy possessed by an object due to its motion. The formula for kinetic energy is \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.
In the scenario of projectile motion, the stone initially has kinetic energy due to its initial speed \( v_0 \). As it moves upwards, its kinetic energy rearranges between its horizontal and vertical motion.
At the top of its path, its vertical kinetic energy is reduced to zero, while the horizontal component \( v_{x} \) maintains kinetic energy. The conserved nature of kinetic energy in the absence of non-conservative forces, like friction or air resistance, is key in analyzing projectile motion.

Potential Energy

Potential energy (PE) depends on the position of an object within a gravitational field. For an object of height \( h \) above a reference point, it is given by the equation \( PE = mgh \).
At the launch, the stone's potential energy is zero at the reference level. However, when the stone reaches its maximum height, the potential energy is at its peak, as all the initial energy has transformed.
The gain in height equates to the conversion of some initial kinetic energy to potential energy. Using energy conservation, we find the total height \( h \) reached by solving \( h = \frac{v_0^2 \sin^2 \theta}{2g} \), illustrating energy transformation from kinetic to potential energy.

Mechanics

Mechanics, a core branch of physics, deals with the motion of objects and the forces acting upon them. It is divided into classical mechanics, which includes the study of projectile motion.
This problem of projecting a stone explores the principles of classical mechanics by analyzing forces, velocities, and energy types. It highlights the conservation laws, which state that energy cannot be created or destroyed but only transformed from one form to another.
The principles of mechanics enable us to use formulas related to kinetic and potential energy effectively to predict heights, velocities, and trajectories of projectiles.

Physics Education

Physics education aims to introduce learners to concepts that explain the natural world. Using real-world problems, such as projectile motion, makes complex theories more engaging and understandable.
When students solve problems involving conservation of energy, they learn to apply theoretical knowledge in practical scenarios, enhancing their problem-solving skills. Understanding these foundational topics prepares learners for more advanced studies in physics and other scientific fields.
Educational platforms often offer step-by-step solutions to exercises, helping to bridge the gap between theory and practical application. Thorough explanations of each step can help clarify the interconnectedness of concepts like kinetic and potential energy within the broader context of physics.