Hillary kicks a ball so that it is in the air for 3.2 s. The ball lands down field 56 m from its starting point.
-Determine how high the ball goes.
-Determine the y coordinate of the ball when its x coordinate is 45 m.
To determine the height the ball reaches, we can use the equation of motion for vertical motion, which is given by:
h = v₀t + (1/2)gt²
where:
h is the height reached by the ball,
v₀ is the initial vertical velocity (which is typically 0 when the ball is kicked),
t is the time the ball is in the air,
and g is the acceleration due to gravity (which is approximately 9.8 m/s²).
Given that the ball is in the air for 3.2 s, we can substitute this value into the equation to find the height:
h = 0 + (1/2)(9.8)(3.2)²
= 0 + (1/2)(9.8)(10.24)
= 0 + (4.9)(10.24)
= 50.176 m
Therefore, the ball reaches a height of approximately 50.176 meters.
To determine the y coordinate of the ball when its x coordinate is 45 m, we need to consider the horizontal motion of the ball. Assuming there is no air resistance, the horizontal motion is uniform and can be described using the equation:
x = v₀x * t
where:
x is the horizontal distance traveled by the ball,
v₀x is the initial horizontal velocity,
and t is the time the ball is in the air.
In this case, we need to find the value of t when x = 45 m. We can rearrange the equation as follows:
t = x / v₀x
To find v₀x, we can use the fact that the horizontal motion is uniform, so the initial horizontal velocity remains constant throughout the ball's flight. Therefore, we can use the formula:
v₀x = d / t
where:
d is the horizontal distance traveled by the ball,
and t is the time the ball is in the air.
Given that the ball lands down field 56 m from its starting point and is in the air for 3.2 s, we can substitute these values into the equation to find v₀x:
v₀x = 56 / 3.2
= 17.5 m/s (rounded to one decimal place)
Now, we can substitute the values of x = 45 m and v₀x = 17.5 m/s into the equation for t to find the time when the x coordinate is 45 m:
t = 45 / 17.5
≈ 2.57 s (rounded to two decimal places)
Therefore, when the x coordinate is 45 m, the y coordinate of the ball can be determined by substituting the value of t into the equation for vertical motion:
h = v₀y * t + (1/2)gt²
Given that the initial vertical velocity v₀y is typically 0 when the ball is kicked, and the acceleration due to gravity g is approximately 9.8 m/s², we can calculate the y coordinate:
h = 0 * 2.57 + (1/2)(9.8)(2.57)²
≈ 33.35 m (rounded to two decimal places)
Therefore, when the x coordinate is 45 m, the y coordinate of the ball is approximately 33.35 meters.