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From what height must you throw a baseball horizontally to have it reach a final velocity of 38.5 m/s in the vertical direction?

Answer :

Final answer:

To find the height from which the baseball must be thrown to reach a final velocity of 38.5 m/s in the vertical direction, we can use the principle of conservation of energy.


Explanation:

To find the height from which the baseball must be thrown, we can use the principle of conservation of energy. The final velocity in the vertical direction is given as 38.5 m/s. This final velocity is achieved when the gravitational potential energy of the ball is converted into kinetic energy. The gravitational potential energy can be calculated using the formula: PE = mgh, 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. Therefore, we can rearrange the formula to solve for h: h = PE / (mg).

Since we want the ball to reach a final velocity of 38.5 m/s, its initial velocity must be zero. Therefore, the kinetic energy is also zero initially, and all the gravitational potential energy is converted into kinetic energy. Thus, we can equate the gravitational potential energy to the kinetic energy: mgh = 1/2 mv², where v is the final velocity. Plugging in the given values, we have: mg(h) = 1/2 m(38.5)². Canceling out the mass, we get: gh = 1/2 (38.5)². Solving for h, we find:

h = (1/2)(38.5)² / g


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Rewritten by : Barada

Final answer:

Using the kinematic equations for projectile motion, specifically for an object in free fall, the height from which a baseball needs to be thrown horizontally to reach a vertical velocity of 38.5 m/s is approximately 76 meters.

Explanation:

To determine the height from which a baseball must be thrown horizontally to reach a vertical velocity of 38.5 m/s, we can use the kinematic equations for projectile motion, specifically the motion relating to free fall under gravity. In this case, we're focused on the vertical component of the motion since the baseball is thrown horizontally (initial vertical velocity is zero).

The formula we use to find the final vertical velocity is:


v2 = u2 + 2as

Where:


  • v is the final velocity in the vertical direction (38.5 m/s)

  • u is the initial velocity (0 m/s as the throw is horizontal)

  • a is the acceleration due to gravity (9.8 m/s2)

  • s is the displacement in the vertical direction (the height from which the ball is thrown)

Rearranging the equation to solve for s:


s = v2 / (2a)

Substitute the known values:


s = (38.5 m/s)2 / (2 × 9.8 m/s2)

Calculate s:


s = 76.07 m

The baseball must be thrown from a height of approximately 76 meters to reach a final vertical velocity of 38.5 m/s.

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