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Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same...

Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground.
Rank from largest to smallest. To rank items as equivalent, overlap them.
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Answer #1
Concepts and reason

The concept required to solve this problem is conservation of energy.

Initially, calculate the kinetic and potential energy of the balls. Finally, rank the baseballs based on their respective final velocities.

Fundamentals

The law of conservation of energy states that the gain in the kinetic energy is equal to the loss in the potential energy.

The expression is as follows:

UiUf=KfKi{U_i} - {U_f} = {K_f} - {K_i}

Here, UiandUf{U_i}{\rm{ and }}{U_f} is the initial and final potential energy respectively and KiandKf{K_i}{\rm{ and }}{K_f} is the initial and final kinetic energy respectively.

The expression foe the kinetic energy is as follows:

T=12mv2T = \frac{1}{2}m{v^2}

Here, m is the mass of the object and v is the velocity of an object.

The expression for the potential energy is as follows:

U=mghU = mgh

Here, g is the acceleration due to gravity and h is the height.

The longer the ball is in the air, the longer it will take to hit the ground. And, larger the final velocity will be when the ball hits the ground.

The balls are thrown from the same height. Thus, the potential energy will be the same for all the balls. Also, when the ball reaches the ground the change in potential energy would also be the same.

According, to the conservation of energy the loss in potential energy is equal to the gain in kinetic energy. Hence the change in potential energy for the balls would be equal to the change in kinetic energy. Also, since the change in potential energy is the same for all the balls the change in kinetic energy will also be the same.

Further since the initial speed of all the balls are same, the final speed will also be the same.

Hence the speed of all the baseballs as they hit the ground would be the same.

Let all the baseballs be thrown from a height hh with an initial velocity uu .

For case1,

The loss in potential energy is,

ΔU1=mgh\Delta {U_1} = mgh

The gain in kinetic energy is,

ΔK1=12m1v1212m1u2\Delta {K_1} = \frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}

Apply the law of conservation of energy.

ΔU1=ΔK1\Delta {U_1} = \Delta {K_1}

Substitute mghmgh for ΔU1\Delta {U_1} and 12m1v1212m1u2\frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}

m1gh=12m1v1212m1u2v1=u2+2gh\begin{array}{c}\\{m_1}gh = \frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}\\\\{v_1} = \sqrt {{u^2} + 2gh} \\\end{array}

Here, v1{v_1} is the final velocity and m1{m_1} is the mass of the baseball.

Similarly, for case2,

The final velocity of baseball 2 will be,

v2=u2+2gh{v_2} = \sqrt {{u^2} + 2gh}

The final velocity of baseball 3 will be,

v3=u2+2gh{v_3} = \sqrt {{u^2} + 2gh}

The final velocity of baseball 4 will be,

v4=u2+2gh{v_4} = \sqrt {{u^2} + 2gh}

The final velocity of baseball 5 will be,

v5=u2+2gh{v_5} = \sqrt {{u^2} + 2gh}

The final velocity of baseball 6 will be,

v6=u2+2gh{v_6} = \sqrt {{u^2} + 2gh}

From the expression for v1,v2,v3,v4,v5andv6{v_1},{v_2},{v_3},{v_4},{v_5}{\rm{ and }}{v_6} it is clear that v1=v2=v3=v4=v5=v6{v_1} = {v_2} = {v_3} = {v_4} = {v_5} = {v_6} .

Ans:

The ranking of the six baseballs is v1=v2=v3=v4=v5=v6{v_1} = {v_2} = {v_3} = {v_4} = {v_5} = {v_6} .

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