five problems designed to test students' ability to use fundamental physics equations with the requirement of algebraic transformations

 Here are five problems designed to test students' ability to use fundamental physics equations with the requirement of algebraic transformations:

1. Problem: Projectile Motion A ball is launched from the ground at an angle $𝜃$ with an initial velocity ${𝑣}_0$. It reaches a maximum height $𝐻$.

   • Given: $𝐻$ and $𝜃$
   • Find: The initial velocity ${𝑣}_0$
   • Formula Involved: $𝐻=\frac{{𝑣}_0^2{\sin }^2𝜃}{2𝑔}$

   Solution Steps:

   • Start with the given formula: $𝐻=\frac{{𝑣}_0^2{\sin }^2𝜃}{2𝑔}$
   • Rearrange to solve for ${𝑣}_0$:
     $${𝑣}_0^2=\frac{2𝑔𝐻}{{\sin }^2𝜃}$$
     $${𝑣}_0=\sqrt{\frac{2𝑔𝐻}{{\sin }^2𝜃}}$$

2. Problem: Kinetic and Potential Energy Conversion A block of mass $𝑚$ slides down a frictionless incline of height $ℎ$ starting from rest. Find the speed of the block at the bottom.

   • Given: $𝑚$, $ℎ$
   • Find: The speed $𝑣$ at the bottom
   • Formula Involved: $𝑚𝑔ℎ=\frac{1}{2}𝑚{𝑣}^2$

   Solution Steps:

   • Start with the energy conservation equation: $𝑚𝑔ℎ=\frac{1}{2}𝑚{𝑣}^2$
   • Cancel out $𝑚$ from both sides:
     $$𝑔ℎ=\frac{1}{2}{𝑣}^2$$
     $${𝑣}^2=2𝑔ℎ$$
     $$𝑣=\sqrt{2𝑔ℎ}$$

3. Problem: Uniform Circular Motion An object is moving in a circle of radius $𝑟$ with a constant speed $𝑣$. Find the centripetal force ${𝐹}_{𝑐}$ acting on the object.

   • Given: $𝑚$, $𝑣$, $𝑟$
   • Find: The centripetal force ${𝐹}_{𝑐}$
   • Formula Involved: ${𝐹}_{𝑐}=\frac{𝑚{𝑣}^2}{𝑟}$

   Solution Steps:

   • Use the centripetal force formula: ${𝐹}_{𝑐}=\frac{𝑚{𝑣}^2}{𝑟}$
   • This problem is straightforward and requires just one algebraic step, but students can be asked to first derive ${𝑣}^2$ if given $𝑇$ (the period):
     $$𝑣=\frac{2𝜋𝑟}{𝑇}$$
     $${𝑣}^2={\left(\frac{2𝜋𝑟}{𝑇}\right)}^2=\frac{4{𝜋}^2{𝑟}^2}{{𝑇}^2}$$
     $${𝐹}_{𝑐}=𝑚\cdot \frac{4{𝜋}^2{𝑟}^2}{𝑟{𝑇}^2}=\frac{4{𝜋}^2𝑚𝑟}{{𝑇}^2}$$

4. Problem: Newton’s Second Law with Friction A block of mass $𝑚$ is pushed across a horizontal surface with a constant force $𝐹$. The coefficient of kinetic friction between the block and the surface is ${𝜇}_{𝑘}$. Find the acceleration $𝑎$ of the block.

   • Given: $𝐹$, $𝑚$, ${𝜇}_{𝑘}$
   • Find: The acceleration $𝑎$
   • Formula Involved: $𝐹-{𝐹}_{\text{friction}}=𝑚𝑎$ and ${𝐹}_{\text{friction}}={𝜇}_{𝑘}𝑚𝑔$

   Solution Steps:

   • Start with the force balance: $𝐹-{𝜇}_{𝑘}𝑚𝑔=𝑚𝑎$
   • Rearrange to solve for $𝑎$:
     $$𝑎=\frac{𝐹-{𝜇}_{𝑘}𝑚𝑔}{𝑚}$$
     $$𝑎=\frac{𝐹}{𝑚}-{𝜇}_{𝑘}𝑔$$

5. Problem: Harmonic Oscillator A mass $𝑚$ attached to a spring with spring constant $𝑘$ oscillates with an amplitude $𝐴$. Find the maximum speed ${𝑣}_{\text{max}}$ of the mass.

   • Given: $𝑚$, $𝑘$, $𝐴$
   • Find: The maximum speed ${𝑣}_{\text{max}}$
   • Formula Involved: ${𝐸}_{\text{total}}=\frac{1}{2}𝑘{𝐴}^2=\frac{1}{2}𝑚{𝑣}_{\text{max}}^2$

   Solution Steps:

   • Start with the energy conservation formula: $\frac{1}{2}𝑘{𝐴}^2=\frac{1}{2}𝑚{𝑣}_{\text{max}}^2$
   • Cancel out the $\frac{1}{2}$ terms:
     $$𝑘{𝐴}^2=𝑚{𝑣}_{\text{max}}^2$$
     $${𝑣}_{\text{max}}=𝐴\sqrt{\frac{𝑘}{𝑚}}$$

These problems are designed to make students use their algebraic skills to manipulate the given equations before arriving at the final answer.