A football is punted at an angle of 44.6 with the horizontal. if it stays in the air for 3.6 seconds, what was the initial velocity of the football?
The vertical component of the velocity is
Vy = V cos 44.6 = .712V
The height of the ball in feet is given by
h(t) = (.712V)t - 16t^2
So, if h(t) = 0 when t=3.6,
.712V*3.6 - 16*3.6^2 = 0
V = 80.9 ft/s
Guy's a pretty decent kicker, since the average kickoff speed is about 50 mph = 73 ft/s
To find the initial velocity of the football, we can use the following kinematic equation:
y = y0 + v0y * t + (1/2) * a * t^2
Where:
- y is the vertical displacement of the football (which we can assume to be zero since the football is punted horizontally)
- y0 is the initial vertical position (also zero in this case)
- v0y is the initial vertical velocity
- t is the time the football stays in the air
- a is the acceleration due to gravity (-9.8 m/s^2)
Since the football is punted horizontally, there is no initial vertical velocity (v0y = 0). The equation then simplifies to:
y = (1/2) * a * t^2
Since y = 0, we have:
0 = (1/2) * a * t^2
Solving for a * t^2 gives:
a * t^2 = 0
Since a and t are nonzero, the only way for a * t^2 to be zero is if a = 0.
Therefore, the acceleration in the vertical direction is zero, which means the football is moving horizontally at a constant velocity.
Hence, the initial velocity of the football is the same as the final velocity in the horizontal direction. We need to find the horizontal component of the initial velocity (v0x).
We can use the following formula to find v0x:
v0x = v * cos(θ)
Where:
- v0x is the horizontal component of the initial velocity
- v is the initial velocity of the football
- θ is the angle at which the football was punted (44.6 degrees in this case)
Substituting the given values:
v0x = v * cos(44.6)
Now, we can solve for v by rearranging the formula:
v = v0x / cos(44.6)
To obtain the value of v0x, we need to know the horizontal distance covered by the football during the 3.6 seconds it stayed in the air.
Can you provide the horizontal distance covered by the football?