Calculating Work Done Using Work-Energy Theorem

The work required to change an object's speed is equal to the change in its kinetic energy. This is known as the Work-Energy Theorem.

The formula for kinetic energy (KE) is given by:

\(KE = \frac{1}{2}mv^2\)

Where:

• \(m\) is the mass of the object.
• \(v\) is the speed of the object.

The work done (\(W\)) is the difference between the final kinetic energy (\(KE_f\)) and the initial kinetic energy (\(KE_i\)):

\(W = \Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\)

Applying the Work-Energy Theorem

Given values:

• Mass (\(m\)): 0.4 kg
• Initial speed (\(v_i\)): 1 m/s
• Final speed (\(v_f\)): 3 m/s

Step 1: Calculate the Initial Kinetic Energy (\(KE_i\))

\(KE_i = \frac{1}{2} \times 0.4 \text{ kg} \times (1 \text{ m/s})^2\)

\(KE_i = \frac{1}{2} \times 0.4 \times 1\)

\(KE_i = 0.2 \text{ J}\)

Step 2: Calculate the Final Kinetic Energy (\(KE_f\))

\(KE_f = \frac{1}{2} \times 0.4 \text{ kg} \times (3 \text{ m/s})^2\)

\(KE_f = \frac{1}{2} \times 0.4 \times 9\)

\(KE_f = 0.2 \times 9\)

\(KE_f = 1.8 \text{ J}\)

Step 3: Calculate the Work Done (\(W\))

\(W = KE_f - KE_i\)

\(W = 1.8 \text{ J} - 0.2 \text{ J}\)

\(W = 1.6 \text{ J}\)

Conclusion

The work required to increase the speed of the ball is 1.6 Joules.