Final answer:
The kinetic energy equation shows that Jake and Harry already have the same kinetic energy when Jake's speed is four times greater and his mass is half that of Harry's. Therefore, Harry does not need to change his speed, and the question contains a conceptual misunderstanding.
Step-by-step explanation:
The subject of this question is Physics, specifically focusing on the concept of kinetic energy, which is commonly included in high school physics curricula. The kinetic energy (KE) of an object is given by the equation KE = 1/2 m v^2, where m is the mass of the object and v is its velocity. To find out by how much Harry needs to increase his speed to have the same kinetic energy as Jake, we set their kinetic energies equal to each other.
- Let's denote Jake's mass as m, Harry's mass as 2m (since Jake's mass is half that of Harry's), Jake's speed as 4v and Harry's original speed as 3 m/s.
- Jake's kinetic energy is KEJake = 1/2 m (4v)^2.
- Harry's kinetic energy is KEHarry = 1/2 (2m) v^2.
- Setting them equal: 1/2 m (4v)^2 = 1/2 (2m) v^2, which simplifies to 8mv^2 = 2mv^2. The m cancels out, and we are left with 8v^2 = 2v^2.
Now, if we substitute Harry's original speed into the equation for his kinetic energy, we can solve for the new speed v he needs:
- Harry's original kinetic energy is KEHarry, original = 1/2 (2m) (3 m/s)^2 = 9m J.
- For Harry to have the same kinetic energy as Jake: 18m J = 1/2 (2m) v^2, which simplifies to 9 = v^2.
- Thus, v is 3 m/s.
- Harry needs to speed up by 3 m/s - 3 m/s = 0 m/s, which might seem counterintuitive.
However, in reality, if Jake's speed were truly 4 times greater than Harry's and his mass were half that of Harry's, their kinetic energies would already be the same due to the fact that kinetic energy is directly proportional to both the square of the speed and the mass of the object. Therefore, there's no need for Harry to speed up to match Jake's kinetic energy, which is consistent with the equation we solved.