A football is thrown directly toward a receiver with an initial speed of 15.8 m/s at an angle of 17° above the horizontal. At that instant, the receiver is 18 m from the quarterback. In what direction and with what constant speed should the receiver run to catch the football at the level at which it was thrown?
To solve this problem, we need to break it down into horizontal and vertical components.
First, let's find the time it takes for the football to reach the receiver. We can use the vertical component for this calculation.
Vertical motion:
- Initial vertical velocity (Vy) = V * sin(θ) = 15.8 m/s * sin(17°) = 4.45 m/s
- Acceleration due to gravity (g) = 9.8 m/s²
Using the equation: Δy = Vy * t + 0.5 * g * t² (where Δy is the vertical displacement),
we can substitute Δy = 0 (as the receiver catches the ball at the same level where it was thrown) and solve for time (t).
0 = 4.45 m/s * t + 0.5 * 9.8 m/s² * t²
0 = 4.45t + 4.9t²
4.9t² + 4.45t = 0
t(4.9t + 4.45) = 0
This equation has two solutions: t = 0 (which is not relevant in this context) and 4.45t + 4.9 = 0.
Solving 4.45t + 4.9 = 0, we get t ≈ -1.10s and t ≈ -0.45s. Since time cannot be negative in this case, we discard the negative solution.
Therefore, the time it takes for the football to reach the receiver is approximately t = 0.45s.
Now, let's find the horizontal distance covered by the football during this time interval.
Horizontal motion:
- Initial horizontal velocity (Vx) = V * cos(θ) = 15.8 m/s * cos(17°) = 15.14 m/s
Using the equation: Δx = Vx * t (where Δx is the horizontal displacement),
we can substitute Vx = 15.14 m/s and t = 0.45s and solve for Δx.
Δx = 15.14 m/s * 0.45s ≈ 6.81m
The football covers a horizontal distance of approximately 6.81m during this time interval.
Now, let's determine the direction and speed the receiver should run to catch the football at the same height it was thrown.
The receiver needs to cover the remaining horizontal distance (18m - 6.81m = 11.19m) during the same time interval (0.45s) in order to meet the football at the same level. Therefore, we can calculate the speed the receiver needs to run.
Speed = Distance / Time
Speed = 11.19m / 0.45s ≈ 24.87 m/s
The receiver should run in the direction of the football (at the angle above the horizontal it was thrown) with a constant speed of approximately 24.87 m/s to catch the football at the level it was thrown.
To solve this problem, we need to break it down into two components: the horizontal component and the vertical component.
First, let's find the time it takes for the football to reach the receiver. We can use the horizontal component of the motion since there is no acceleration in that direction. The equation we can use is:
distance = speed * time
The distance here is the 18 meters between the quarterback and the receiver. The speed is the horizontal component of the initial speed, which is calculated by multiplying the initial speed by the cosine of the angle (17°):
18 meters = (15.8 m/s) * cos(17°) * time
Now we can solve for time:
time = 18 meters / [(15.8 m/s) * cos(17°)]
Next, let's find the vertical component of the motion at the time it takes for the football to reach the receiver. We can use the equation:
vertical displacement = (initial vertical velocity * time) + (0.5 * acceleration * time^2)
Since the receiver wants to catch the ball at the same level at which it was thrown, the vertical displacement will be zero:
0 = (initial vertical velocity * time) + (0.5 * acceleration * time^2)
The initial vertical velocity is calculated by multiplying the initial speed by the sine of the angle (17°):
0 = (15.8 m/s) * sin(17°) * time + (0.5 * (-9.8 m/s^2) * time^2)
Now we can solve this equation to find the time it takes for the ball to reach the receiver:
0 = (15.8 m/s) * sin(17°) * time - 4.9 m/s^2 * time^2
This equation is a quadratic equation with respect to time. Solve for time using your preferred method (factoring, quadratic formula, etc.).
Once you have the time it takes for the ball to reach the receiver, you can find the constant speed the receiver needs to run at by dividing the horizontal distance traveled by the time:
receiver's constant speed = 18 meters / time
To find the direction the receiver needs to run, you can use trigonometry. The angle of the receiver's motion with respect to the horizontal is given by:
receiver's angle = tan^(-1)[(initial vertical velocity - gravity * time) / (initial horizontal velocity)]
Substitute the values we calculated earlier to find the receiver's angle. This will give you the direction in which the receiver needs to run to catch the football at the level it was thrown.