Conservation of Mechanical Energy (Edexcel A-Level Further Mathematics): Revision Notes

Example 1: Object Falling Freely Under Gravity

Problem

A ball of mass 2 kg is dropped from a height of 10 m.

Ignoring air resistance, find:

1. The speed of the ball just before it hits the ground.
2. The kinetic energy at that point.

Part 1: Speed at the Ground

Step 1: Use conservation of mechanical energy:

Step 2: At the top:

Step 3: At the bottom:

Step 4: Equate total energy at the top and bottom:

Step 5: Solve for $v$:

Part 2: Kinetic Energy at the Ground

Step 1: Use the kinetic energy formula:

Step 2: Substitute $m = 2 \, \text{kg}$ and $v = 14 \, \text{ms}^{-1}$

Final Answer:

Speed: 14 ms⁻¹

Kinetic Energy: 196 J

Example 2: Block Sliding Down a Smooth Incline

Problem

A block of mass 3 kg slides down a smooth incline of length 5 m that makes an angle of 30° with the horizontal.

The block starts from rest at the top. Find:

3. The speed of the block at the bottom.
4. The total mechanical energy at the top and bottom.

Part 1: Speed at the Bottom

Step 1: Use conservation of mechanical energy:

Step 2: At the top:

where $h$ is the height of the incline:

Step 3: At the bottom:

Step 4: Equate total energy:

Step 5: Solve for vv:

Part 2: Total Mechanical Energy

At the top:

At the bottom:

Final Answer:

Speed: 7 ms⁻¹

Total Mechanical Energy: 73.5 J (at both the top and bottom)

Common Mistakes

1. Forgetting to calculate height correctly: Use $h = d \sin \theta$ for inclined planes.
2. Mixing energy types: Ensure you correctly identify initial and final KE and PE.
3. Neglecting conservation principles: Remember that total mechanical energy remains constant if no external work is done.
4. Omitting units: Always express energy in joules (J) and speeds in metres per second (ms⁻¹).
5. Ignoring starting conditions: Include initial velocity in calculations when relevant.

Key Formulas

1. Kinetic Energy:

2. Potential Energy:

3. Conservation of Mechanical Energy:

4. Height on an Incline: