To find the amount of work done by the runner to accelerate from rest to a velocity of 3 m/s, we can use the concept of kinetic energy. The work done on an object is equal to the change in its kinetic energy.
The kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where:
- \( m \) is the mass of the object (in kg),
- \( v \) is the velocity of the object (in m/s).
Initially, the runner is at rest, so the initial velocity (\( v_i \)) is 0 m/s. The final velocity (\( v_f \)) is 3 m/s. The mass \( m \) of the runner is 55 kg.
- Calculate the initial kinetic energy (\( KE_i \)):
\[ KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} \times 55 , \text{kg} \times (0 , \text{m/s})^2 = 0 \]
- Calculate the final kinetic energy (\( KE_f \)):
\[ KE_f = \frac{1}{2} m v_f^2 = \frac{1}{2} \times 55 , \text{kg} \times (3 , \text{m/s})^2 = \frac{1}{2} \times 55 \times 9 = \frac{495}{2} = 247.5 , \text{J} \]
- Calculate the work done (\( W \)), which is the change in kinetic energy:
\[ W = KE_f - KE_i = 247.5 , \text{J} - 0 , \text{J} = 247.5 , \text{J} \]
Therefore, the amount of work that the 55 kg runner must do to accelerate from rest to a velocity of 3 m/s is 247.5 joules.
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