A rookie quarterback throws a football with an initial upward velocity component of 15.9 m/s and a horizontal velocity component of 18.0 m/s . Ignore air resistance.
a) How much time is required for the football to reach the highest point of the trajectory?
c) How much time (after it is thrown) is required for the football to return to its original level?
d) How does this compare with the time calculated in part (a).
e) How far has it traveled horizontally during this time?
a) 15.9 / g = 15.9 m/s / 9.8 m/s^2
c) time up equals time down
... twice a)
e) time from c) times 18.0 m/s
To answer these questions, we can use the kinematic equations of motion to analyze the motion of the football. Let's break down the problem step by step:
a) How much time is required for the football to reach the highest point of the trajectory?
To find the time it takes for the ball to reach the highest point, we need to consider the vertical motion. We can use the kinematic equation:
v_f = v_i + a * t
Where:
v_f is the final vertical velocity (which is 0 m/s at the highest point),
v_i is the initial vertical velocity (15.9 m/s),
a is the vertical acceleration (which is the acceleration due to gravity, approximately -9.8 m/s²),
and t is the time.
We can rearrange the equation to solve for t:
0 = 15.9 m/s - 9.8 m/s² * t
Simplifying the equation:
9.8 m/s² * t = 15.9 m/s
t = 15.9 m/s / 9.8 m/s²
Solving this equation, we get:
t ≈ 1.63 seconds
So, it takes approximately 1.63 seconds for the football to reach the highest point of its trajectory.
c) How much time (after it is thrown) is required for the football to return to its original level?
The total time for the football to go up and come back down is twice the time calculated in part (a). Therefore:
Total time = 2 * (1.63 seconds)
Total time ≈ 3.26 seconds
So, it takes approximately 3.26 seconds for the football to return to its original level after being thrown.
d) How does this compare with the time calculated in part (a)?
The time calculated in part (a) is the time it takes for the football to reach the highest point. Comparing the two values:
Time to reach the highest point ≈ 1.63 seconds
Total time to return to the original level ≈ 3.26 seconds
We can observe that the total time to return to the original level is twice the time to reach the highest point. This is because the ball takes the same amount of time to go up as it takes to come back down.
e) How far has it traveled horizontally during this time?
To find the horizontal distance traveled, we can use the horizontal velocity (18.0 m/s) and the time calculated in part (c) (3.26 seconds):
Distance = Velocity * Time
Distance = 18.0 m/s * 3.26 seconds
Distance ≈ 58.68 meters
So, the football has traveled approximately 58.68 meters horizontally during the time it takes to return to its original level.