Step 1: Use the conservation of energy
When an object moves in Earth's gravity, its total energy (kinetic plus potential) stays the same if no energy is lost.
Step 2: Write the energy equation
The equation is:
‌mVi2−‌=‌mVf2+0Here,
Vi is the speed at the start, and
Vf is the speed far away from Earth (where gravity is almost zero).
Step 3: Substitute the initial speed
The body is thrown with speed
Vi=√5Ve, where
Ve is escape velocity.
Step 4: Plug in values and simplify
‌‌m(√5Ve)2−‌=‌mVf2‌⇒‌mVe2−gRm=‌mVf2‌⇒‌Ve2−gR=‌Vf2Step 5: Express
gR in terms of
VeRemember,
Ve=√2gR. Square both sides to get
Ve2=2gR.
So,
gR=‌.
Step 6: Substitute and solve for
VfPlug
gR=‌ into the previous equation:
‌‌Ve2−‌=‌Vf2‌(5−1)‌=‌Vf2‌4‌=‌Vf2Multiply both sides by 2 :
4Ve2=Vf2Take square root:
Vf=2Ve
