The Sketch Below Shows a Bowling Ball Return at a Bowling

Question

Examlex · Accepted Answer

The Answer is To estimate the required speed of the bowling ball at point 1 so that it reaches point 2 with a speed of

2 \mathrm {~m} / \mathrm { s }

, we need to apply the principle of conservation of mechanical energy, assuming there is no friction or air resistance.

The mechanical energy of the ball consists of its kinetic energy (KE) and potential energy (PE). At point 1, the ball will have a certain amount of kinetic energy due to its speed and potential energy due to its height above some reference level. At point 2, the ball will also have kinetic and potential energy.

Let's denote:
-

v_1

as the speed of the ball at point 1,
-

v_2 = 2 \mathrm {~m} / \mathrm { s }

as the speed of the ball at point 2,
-

h_1

as the height of the ball at point 1,
-

h_2

as the height of the ball at point 2,
-

m

as the mass of the ball (which will cancel out in the end),
-

g

as the acceleration due to gravity (

9.81 \mathrm {~m} / \mathrm { s }^2

).

The conservation of mechanical energy states that the total mechanical energy at point 1 is equal to the total mechanical energy at point 2:

KE_1 + PE_1 = KE_2 + PE_2

\frac{1}{2} m v_1^2 + m g h_1 = \frac{1}{2} m v_2^2 + m g h_2

Since the mass of the ball

m

appears in every term, it can be canceled out:

\frac{1}{2} v_1^2 + g h_1 = \frac{1}{2} v_2^2 + g h_2

Now, we need to solve for

v_1

:

\frac{1}{2} v_1^2 = \frac{1}{2} v_2^2 + g h_2 - g h_1

v_1^2 = v_2^2 + 2g (h_2 - h_1)

v_1 = \sqrt{v_2^2 + 2g (h_2 - h_1)}

However, the height

h_2

is not provided in the question. To proceed, we need the value of

h_2 - h_1

. Assuming

h_2 - h_1

is given or can be measured, you can plug in the values to find

v_1

.

For example, if

h_2 - h_1 = 1 \mathrm {~m}

, then:

v_1 = \sqrt{(2 \mathrm {~m} / \mathrm { s })^2 + 2 \cdot 9.81 \mathrm {~m} / \mathrm { s }^2 \cdot 1 \mathrm {~m}}

v_1 = \sqrt{4 + 19.62}

v_1 = \sqrt{23.62}

v_1 \approx 4.86 \mathrm {~m} / \mathrm { s }

So, if the height difference is 1 meter, the ball would need to be traveling at approximately

4.86 \mathrm {~m} / \mathrm { s }

at point 1 to reach point 2 with a speed of

2 \mathrm {~m} / \mathrm { s }

. If the height difference is different, you would need to adjust the calculation accordingly.

The Sketch Below Shows a Bowling Ball Return at a Bowling

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