Standing at the edge of a cliff 240 m tall, you throw two balls into the air – one directly upwards at 5 m/s and another directly downwards at -5 m/s. Use conservation of energy to show that they will have the same final speed at the bottom.
The 1st ball:
PE+KE₀=KE
mgH+mv₀²/2 =mv1²/2
v1=sqrt(2gH+ v₀²)
The 2nd ball:
upwards motion
KE1=PE1
mv₀²/2 =mgh
h= v₀²/2g.
downwards motion:
PE2=KE2
mg(H+h) = mv2²/2
v2=sqrt{ 2g(H+h)}=
= sqrt{ 2g(H+ v₀²/2g )}=
= sqrt {2gH + v₀²}
=> v2=v1
Could you break it down a little further?
To use the principle of conservation of energy to show that the two balls will have the same final speed at the bottom of the cliff, we need to analyze the energy transformations that occur.
Let's break down the problem into steps:
Step 1: Potential Energy at the Top
Both balls start at the same height, which is the top of the 240 m cliff. The potential energy at the top for both balls is given by:
PE(top) = m * g * h
where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the cliff (240 m).
Step 2: Initial Kinetic Energy
The two balls are thrown with different velocities, one upwards at 5 m/s and the other downwards at -5 m/s. The initial kinetic energy for both balls is given by:
KE(initial) = 1/2 * m * v²
where v is the initial velocity of the ball.
Step 3: Potential Energy at the Bottom
As the balls reach the bottom of the cliff, they will have converted all their potential energy into kinetic energy. The potential energy at the bottom is zero, as the height is zero.
PE(bottom) = 0
Step 4: Final Kinetic Energy
The final kinetic energy at the bottom is the total energy that the ball possesses just before striking the ground.
KE(final) = 1/2 * m * v₂²
where v₂ is the final velocity of the ball at the bottom.
Now let's apply the conservation of energy principle:
According to the principle of conservation of energy, the total energy at the top (potential energy) should be equal to the total energy at the bottom (kinetic energy). Mathematically:
PE(top) + KE(initial) = PE(bottom) + KE(final)
Substituting the equations from Steps 1, 2, and 3:
m * g * h + 1/2 * m * v₁² = 0 + 1/2 * m * v₂²
We can cancel out the mass, m, from both sides of the equation:
g * h + 1/2 * v₁² = 1/2 * v₂²
Now, let's plug in the given values:
g = 9.8 m/s² (acceleration due to gravity)
h = 240 m (height of the cliff)
v₁ = 5 m/s (initial velocity upwards)
9.8 * 240 + 1/2 * 5² = 1/2 * v₂²
2352 + 12.5 = 1/2 * v₂²
2364.5 = 1/2 * v₂²
Divide both sides by 1/2:
4729 = v₂²
Finally, take the square root of both sides:
v₂ = sqrt(4729) ≈ 68.79 m/s
Therefore, both balls will have the same final speed at the bottom of approximately 68.79 m/s.