To determine the work done on the runner as they increase their speed, we need to calculate the change in their kinetic energy.
Kinetic energy (KE) is given by the formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where [tex]\( m \)[/tex] is the mass and [tex]\( v \)[/tex] is the velocity.
### Step-by-Step Solution:
1. Calculate the initial kinetic energy:
[tex]\[ \text{Initial KE} = \frac{1}{2} \times 60.0 \, \text{kg} \times (6.00 \, \text{m/s})^2 \][/tex]
[tex]\[ \text{Initial KE} = \frac{1}{2} \times 60.0 \times 36 \][/tex]
[tex]\[ \text{Initial KE} = 30.0 \times 36 \][/tex]
[tex]\[ \text{Initial KE} = 1080.0 \, \text{J} \][/tex]
2. Calculate the final kinetic energy:
[tex]\[ \text{Final KE} = \frac{1}{2} \times 60.0 \, \text{kg} \times (9.00 \, \text{m/s})^2 \][/tex]
[tex]\[ \text{Final KE} = \frac{1}{2} \times 60.0 \times 81 \][/tex]
[tex]\[ \text{Final KE} = 30.0 \times 81 \][/tex]
[tex]\[ \text{Final KE} = 2430.0 \, \text{J} \][/tex]
3. Calculate the work done:
The work done is equal to the change in kinetic energy.
[tex]\[ \text{Work Done} = \text{Final KE} - \text{Initial KE} \][/tex]
[tex]\[ \text{Work Done} = 2430.0 \, \text{J} - 1080.0 \, \text{J} \][/tex]
[tex]\[ \text{Work Done} = 1350.0 \, \text{J} \][/tex]
Thus, the work done on the runner as they speed up from [tex]\( 6.00 \, \text{m/s} \)[/tex] to [tex]\( 9.00 \, \text{m/s} \)[/tex] is [tex]\( 1350 \, \text{J} \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{1350 \, \text{J}} \][/tex]
