A quarterback throws a football toward a receiver with an initial speed of 21 m/s at an angle of 30° above the horizontal. At that instant, the receiver is 12 m from the quarterback. In what direction and with what constant speed should the receiver run in order to catch the football at the level at which it was thrown?
Using d = (V^2/g)sin2µ
the distance the football travels derives from d = (21^2/9.8)sin(60) = 38.97 m.
The time it will take the ball to reach the receivers hands derives from Vf = Vo - gt or 0 = 21sin60 - 9.8t making t = 1.85 sec.
Consequently, the receiver should run away from the quarterback at a speed of (38.97 - 12)/1.85 = 14.58m/s.
that is the speed I originally got, but it is supposedly wrong.
To catch the football at the level at which it was thrown, the receiver needs to run in a certain direction with a certain speed. Let's break it down step by step:
Step 1: Find the horizontal and vertical components of the initial velocity of the football.
The horizontal component (Vx) can be found using the formula Vx = V * cos(θ), where V is the initial speed of the football and θ is the angle above the horizontal.
Thus, Vx = 21 m/s * cos(30°) = 21 m/s * √3/2 = 21√3/2 ≈ 18.2 m/s.
The vertical component (Vy) can be found using the formula Vy = V * sin(θ), where V is the initial speed of the football and θ is the angle above the horizontal.
Thus, Vy = 21 m/s * sin(30°) = 21 m/s * 1/2 = 10.5 m/s.
Step 2: Calculate the time it takes for the football to reach the receiver.
The horizontal distance between the quarterback and receiver is 12 m. Using this distance and the horizontal component of velocity, we can calculate the time (t) using the formula d = Vx * t, where d is the horizontal distance.
Thus, 12 m = (21√3/2 m/s) * t.
Solving for t, we get t = 12 / (21√3/2) sec ≈ 0.45 sec.
Step 3: Determine the vertical velocity of the receiver.
Since the receiver needs to catch the ball at the level at which it was thrown, the initial vertical velocity of the receiver should be the same as the vertical velocity of the ball. Thus, the receiver needs to run upward with a velocity of 10.5 m/s.
Step 4: Calculate the speed at which the receiver should run.
The receiver needs to run in such a way that the vertical position of the receiver matches the vertical position of the football when the football reaches the receiver's horizontal position. Since the receiver needs to cover 12 m horizontally in 0.45 sec, the speed at which the receiver should run can be calculated using the formula v = d / t, where d is the distance and t is the time.
Thus, v = 12 m / 0.45 sec ≈ 26.67 m/s.
Step 5: Determine the direction in which the receiver should run.
The direction in which the receiver should run can be determined using trigonometry. The receiver needs to run with the same angle as the initial direction of the ball, which is 30° above the horizontal.
So, the receiver should run upward with a speed of approximately 26.67 m/s at an angle of 30° above the horizontal to catch the football at the level at which it was thrown.
To solve this problem, we can break it down into two components: the horizontal and vertical motions.
First, let's analyze the horizontal motion. Since there is no acceleration in the horizontal direction (assuming no air resistance), the receiver should run in a straight line towards the point where the football will land. The horizontal distance (12 m) is already given, so the receiver should run directly towards the quarterback.
Next, let's analyze the vertical motion. We need to find the time it takes for the football to reach the receiver's position. We can use the kinematic equation:
$$
y = y_0 + v_0y t - \frac{1}{2}gt^2
$$
where
- $y$ is the vertical displacement (0 m), since the receiver wants to catch the football at the same height it was thrown,
- $y_0$ is the initial vertical displacement (0 m),
- $v_0y$ is the initial vertical velocity component (21 m/s * sin(30°)),
- $t$ is the time,
- $g$ is the acceleration due to gravity (-9.8 m/s^2).
Substituting the values, the equation becomes:
$$
0 = 0 + (21 \sin 30°) t - \frac{1}{2}(9.8)t^2
$$
Simplifying, we have:
$$
(10.5)t - 4.9t^2 = 0
$$
Factoring out t, we get:
$$
t(10.5 - 4.9t) = 0
$$
This equation has two solutions: t = 0 (which we can ignore) and t = 2.14 seconds.
Now that we have the time it takes for the football to reach the receiver's position, we can find the constant speed the receiver needs to run.
The horizontal distance (12 m) divided by the time (2.14 s) gives us the constant speed:
$$
\text{Constant speed} = \frac{12 \text{ m}}{2.14 \text{ s}} = 5.61 \text{ m/s}
$$
Therefore, the receiver needs to run in the direction of the quarterback with a constant speed of 5.61 m/s in order to catch the football at the level it was thrown.