Two brothers are jogging, Josh and Drake. Josh has 1/2 the KE of Drake. Also, he is twice the mass of Drake. Plus, Josh had to increase his speed by 1.35 meters/second to achieve the same kinetic energy of Drake. What is the initial velocity?
JoshKE=1/2DrakeKE
1/2Mj vj^2=1/4 Md Vd^2
1/2 2Md Vj^2=1/4 Md Vd^2
Vj^2=1/4 Vd^2
Vj= 1/2 Vd
but Vj+1.35 yields..
1/2 *2Md(Vj+1.35)^2=1/2 Md Vd^2
2(Vj+1.35)^2=Vd^2
finally
2(Vj+1.35)^2=4Vj^2
Vj+1.35=Vj sqrt2
Vj=1.35/.414=
check all that.
To solve this problem, we can use the formula for kinetic energy (KE):
KE = (1/2) * mass * velocity^2
Let's assume that Drake's initial velocity is v, and Josh's initial velocity is 2v (since Josh is twice the mass of Drake). We are given that Josh had to increase his velocity by 1.35 m/s to achieve the same kinetic energy as Drake.
Using the formula for kinetic energy, we can write the following equations:
(1/2) * mass_Drake * v^2 = (1/2) * mass_Josh * (2v + 1.35)^2
Let's define m_Drake as the mass of Drake, and m_Josh as the mass of Josh.
Now we can substitute the values into the equation:
(1/2) * m_Drake * v^2 = (1/2) * m_Josh * (2v + 1.35)^2
We know that m_Josh = 2 * m_Drake (since Josh is twice the mass of Drake):
(1/2) * m_Drake * v^2 = (1/2) * (2 * m_Drake) * (2v + 1.35)^2
Simplifying the equation:
m_Drake * v^2 = 2 * m_Drake * (2v + 1.35)^2
Let's divide both sides of the equation by m_Drake:
v^2 = 2 * (2v + 1.35)^2
Expanding and simplifying:
v^2 = 2 * (4v^2 + 5.4v + 1.8225)
v^2 = 8v^2 + 10.8v + 3.645
Rearranging the equation:
7v^2 - 10.8v - 3.645 = 0
Now we can use the quadratic formula to solve for v:
v = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 7, b = -10.8, and c = -3.645.
Plugging in the values:
v = (-(-10.8) ± √((-10.8)^2 - 4 * 7 * -3.645)) / (2 * 7)
Simplifying:
v = (10.8 ± √(116.64 + 101.94)) / 14
v = (10.8 ± √(218.58)) / 14
v = (10.8 ± 14.8) / 14
Now we have two possible values for v:
v₁ = (10.8 + 14.8) / 14 = 25.6 / 14 = 1.83 m/s
v₂ = (10.8 - 14.8) / 14 = -4 / 14 = -0.29 m/s
Since velocity cannot be negative, we discard v₂.
Therefore, the initial velocity, v, is approximately 1.83 m/s.