Final answer:
Jack must increase his speed to approximately 16.97 m/s to have the same kinetic energy as Oliver, who runs at 24 m/s.
Step-by-step explanation:
To determine by how much Jack needs to speed up to have the same kinetic energy as Oliver, we can use the formula for kinetic energy, KE = 0.5 × m × v2, where m is mass and v is velocity. Since Oliver's mass is half that of Jack, let's denote Jack's mass as 2m and Oliver's mass as m. Given that Oliver's speed is 8 times Jack's original speed of 3 m/s, Oliver's speed is 8 × 3 m/s = 24 m/s.
Setting the kinetic energies equal to each other, we have:
0.5 × m × (24 m/s)2 = 0.5 × 2m × v2
which simplifies to:
576m = 2mv2
Dividing both sides by 2m, we get:
v2 = 288
Taking the square root of both sides, v = √288 ≈ 16.97 m/s.
Therefore, Jack must speed up to approximately 16.97 m/s to have the same kinetic energy as Oliver.