# A baseball thrown from shortstop position to first base, travels 32 m horizontally, rises 3.0 m and falls 3.0 m. Find the magnitude of the initial velocity?

The key to these projectile questions is to treat the horizontal and vertical components of motion separately and use the fact that they share the same time of flight.

When the baseball has reached its height of 3m it falls under gravity so we can write:

This means the total time of flight will be:

If

Since

Then:

So:

To get

We can use the equation of motion:

This becomes:

Dividing

From which:

This is the angle of launch.

To get the launch velocity

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Use the kinematic equation: ( v_i = \sqrt{\frac{g \cdot d^2}{2 \cdot (h_f - h_i)}} ), where (g) is the acceleration due to gravity (9.8 m/s²), (d) is the horizontal distance (32 m), (h_f) is the final height (3.0 m), and (h_i) is the initial height (0, as it starts from the same level). Calculate (v_i).

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To find the magnitude of the initial velocity of the baseball, you can use the kinematic equations of motion. Since the motion can be divided into horizontal and vertical components, you can analyze each component separately.

For the horizontal motion, the displacement is ( 32 ) meters and the time of flight is the same for both components. Using the equation:

[ \text{Displacement} = \text{Initial velocity} \times \text{Time} ]

For the vertical motion, the displacement is ( 3.0 ) meters upwards and then ( 3.0 ) meters downwards, resulting in a net displacement of ( 0 ) meters. Therefore, the time taken to rise and fall must be equal.

Use the kinematic equation for vertical motion:

[ d = v_{i}t - \frac{1}{2}gt^2 ]

Where ( d = 3.0 ) meters (either upwards or downwards, as the time of ascent equals the time of descent), ( g = 9.8 , \text{m/s}^2 ), and ( v_{i} ) is the initial vertical velocity.

From the horizontal motion, you can find the time of flight. Then, you can substitute this time into the vertical motion equation to find the initial vertical velocity. Finally, you can use the Pythagorean theorem to find the magnitude of the initial velocity, considering both horizontal and vertical components.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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