A baseball is tossed at a steep angle into the air and makes a smooth parabolic path. Its time in the air is t and it reaches a maximum height h. Assume that air resistance is negligible.
a. Show that the height reached by the ball is gt^2/8.
b. If the ball is in the air for 4 seconds, show that the ball reaches a height of 19.6m.
c. If the ball reached the same height as when tossed at some other angle, would the time of flight be the same?
A baseball is tossed at a steep angle into the air and makes a smooth parabolic path. Its time in the air is t and it reaches a maximum height h. Assume that air resistance is negligible.
a. Show that the height reached by the ball is gt^2/8.
b. If the ball is in the air for 4 seconds, show that the ball reaches a height of 19.6m.
c. If the ball reached the same height as when tossed at some other angle, would the time of flight be the same?
a. To show that the height reached by the ball is gt^2/8, we can use the equations of motion for an object in free fall. In this case, the ball is thrown upwards, but we can still use the same equations.
The equation for the vertical position of an object in free fall is given by:
h = (1/2)gt^2 + v0t + h0
Where:
h is the height reached by the ball
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time in the air
v0 is the initial vertical velocity (which is 0 because the ball is thrown upwards)
h0 is the initial height (which is also 0 because we are measuring the height from the starting point)
Plugging in the values:
h = (1/2)gt^2 + 0t + 0
h = gt^2/2
Now, dividing both sides of the equation by 2:
h = gt^2/2
h = gt^2/8
Therefore, the height reached by the ball is gt^2/8.
b. To show that the ball reaches a height of 19.6m when in the air for 4 seconds, we can substitute the values into the equation derived in part a.
h = gt^2/8
Plugging in the values:
h = (9.8 m/s^2)(4 s)^2/8
h = (9.8 m/s^2)(16 s^2)/8
h = 156.8 m^2/s^2/8
h = 19.6 m
Therefore, the ball reaches a height of 19.6m when in the air for 4 seconds.
c. If the ball reached the same height as when tossed at some other angle, the time of flight would not necessarily be the same. The time of flight depends on the initial vertical velocity of the ball. If the ball is thrown at a steeper angle, it will have a greater initial vertical velocity and will therefore stay in the air for a shorter period of time. On the other hand, if the ball is thrown at a less steep angle, it will have a smaller initial vertical velocity and will stay in the air for a longer period of time. So, the time of flight is determined by the initial vertical velocity and not just the height reached.