Question

A soccer ball is kicked from the ground with an initial speed of 19.5 m/s. Determine the ball's maximum height and the time it takes to reach that height.

Answer 1

The soccer ball reaches a maximum height of approximately
`\(h = 19.8 \ \mathrm{m}\) and takes approximately \(t = 2.0 \ \mathrm{s}\) to reach that height.`

Step-by-step explanation:

To determine the maximum height
`(\(h\))` and the time
`(\(t\))` it takes for the soccer ball to reach that height, we can use the kinematic equations of motion. The key equation for this scenario is:

`\[ h = (v_0^2 \sin^2(\theta))/(2g) \]`

where
`\(v_0\)` is the initial speed
`(given as \(19.5 \ \mathrm{m/s}\)), \(\theta\) is the launch angle (assumed to be 90 degrees for vertical motion), and \(g\) is the acceleration due to gravity (\(9.8 \ \mathrm{m/s^2}\)).`

Plugging in the values, we get:

`\[ h = \frac{(19.5 \ \mathrm{m/s})^2 \sin^2(90^\circ)}{2 \cdot 9.8 \ \mathrm{m/s^2}} \]`

Simplifying the equation yields the maximum height
`(\(h\)).`

Next, we can determine the time
`(\(t\))` it takes for the ball to reach this height. Using the kinematic equation:

`\[ t = (v_0 \sin(\theta))/(g) \]`

Substituting the known values:

`\[ t = \frac{19.5 \ \mathrm{m/s} \cdot \sin(90^\circ)}{9.8 \ \mathrm{m/s^2}} \]`

This gives us the time taken to reach the maximum height
`(\(t\)).`Therefore, the soccer ball reaches a maximum height of approximately
`\(19.8 \ \mathrm{m}\) and takes approximately \(2.0 \ \mathrm{s}\) to reach that height.`

Answer 2

The maximum height of the soccer ball is approximately 19.54 meters, and it takes approximately 1.99 seconds to reach that height.

Step-by-step explanation:

To find the maximum height and time taken for a soccer ball kicked with an initial velocity of 19.5 m/s, we need to consider the vertical motion of the ball. Assuming no air resistance, we can use the equations of motion to solve for these values.

The initial vertical velocity of the ball is 19.5 m/s since it is kicked vertically. The acceleration due to gravity, (g), is 9.8 m/s² downwards.

h = (v₀² - v²) / (2g), where v₀ is the initial vertical velocity and v is the final vertical velocity (which is 0 when the ball reaches its maximum height). h = (19.5² - 0) / (2 * 9.8) = 19.5² / (2 * 9.8) = 19.5² / 19.6 ≈ 19.54 m

So, the maximum height reached by the ball is approximately 19.54 meters. t = (v - v₀) / g, where t is the time and v is the final vertical velocity.

t = (0 - 19.5) / 9.8 = -19.5 / 9.8 ≈ -1.99 s

Since time cannot be negative, we take the absolute value of the result, giving us a time of approximately 1.99 seconds.