A quarterback throws a football toward a re-
ceiver with an initial speed of 25 m/s, at an
angle of 29◦ above the horizontal. At that
instant, the receiver is 19 m from the quarter-
back.
The acceleration of gravity is 9.8 m/s2 .
With what constant speed should the re-
ceiver run in order to catch the football at the
level at which it was thrown?
Answer in units of m/s.
To solve this problem, we need to find the constant speed at which the receiver should run to catch the football at the level it was thrown.
Let's break down the information we have:
- Initial speed of the football (v0) = 25 m/s
- Angle above the horizontal (θ) = 29°
- Distance between the receiver and quarterback (d) = 19 m
First, we need to find the time it takes for the football to reach the receiver. We can use the horizontal motion equation:
d = v0 * t * cos(θ)
Rearranging the equation to solve for time (t):
t = d / (v0 * cos(θ))
Plugging in the values:
t = 19 m / (25 m/s * cos(29°))
Now, we can find the vertical distance the football drops during that time using the vertical motion equation:
h = 1/2 * g * t^2
Where g is the acceleration due to gravity, which is 9.8 m/s^2.
Plugging in the values:
h = 1/2 * 9.8 m/s^2 * (19 m / (25 m/s * cos(29°)))^2
Next, we can set up the equation for the vertical motion of the receiver. The receiver's initial vertical velocity is 0 m/s, and the acceleration is also 0 m/s^2 since the receiver is running at a constant speed. The equation becomes:
h = 1/2 * 0 m/s^2 * t^2 + v * t
Since the acceleration is 0 m/s^2, the equation simplifies to:
h = v * t
Rearranging the equation to solve for the constant speed (v) of the receiver:
v = h / t
Plugging in the values:
v = (1/2 * 9.8 m/s^2 * (19 m / (25 m/s * cos(29°)))^2) / (19 m / (25 m/s * cos(29°)))
Calculating this gives us the answer, which is the constant speed at which the receiver should run in order to catch the football at the level it was thrown.