The kinetic energy (KE) of an object can be calculated using the formula:
\[ KE = \frac{1}{2}mv^2 \]
where m is the mass of the object and v is its velocity.
When you are running around a track, your kinetic energy is determined by your speed. Initially, you are running at an initial speed of 5 m/s, which we'll call \( v_i \). Then you increase your speed to 10 m/s, which we'll call \( v_f \).
The initial kinetic energy, \( KE_i \), when you are running at 5 m/s is proportional to the square of your initial speed:
\[ KE_i ∝ (v_i)^2 \]
\[ KE_i ∝ (5)^2 \]
\[ KE_i ∝ 25 \]
After increasing your speed to 10 m/s, your final kinetic energy, \( KE_f \), is proportional to the square of your final speed:
\[ KE_f ∝ (v_f)^2 \]
\[ KE_f ∝ (10)^2 \]
\[ KE_f ∝ 100 \]
To find the factor by which your kinetic energy increased, you would divide your final kinetic energy by your initial kinetic energy:
\[ \text{Factor of Increase} = \frac{KE_f}{KE_i} \]
Putting in the proportionality constants we have:
\[ \text{Factor of Increase} = \frac{100}{25} \]
\[ \text{Factor of Increase} = 4 \]
So, when you increase your speed from 5 m/s to 10 m/s, you increase your kinetic energy by a factor of 4.
