A football is thrown toward a receiver with
an initial speed of 15.4 m/s at an angle of
38.7
◦
above the horizontal. At that instant,
the receiver is 17.7 m from the quarterback.
The acceleration of gravity is 9.81 m/s
2
.
With what constant speed should the receiver run to catch the football at the level at
which it was thrown?
Answer in units of m/s
Vo = 15.4m/s[38.7o]
Xo = 15.4*cos38.7 = 12 m/s.
Yo = 15.4*sin38.7 = 9.63 m/s.
Range = Vo^2*sin(2A)/g
Range = 15.4^2*sin(77.4)/9.8 = 23.62 m.
Range = Xo * T = 23.62 m.
Range = 12 * T = 23.62
T = 1.97 s. in air.
d=Vr * T = 23.62-17.7 = 5.92 m. to run.
Vr * 1.97 = 5.92
Vr = 3.0 m/s. = Speed of the receiver.
To find the constant speed at which the receiver should run to catch the football at the level it was thrown, we need to consider the horizontal and vertical components of the motion separately.
First, let's find the time it takes for the football to reach the receiver. We can use the horizontal component of the velocity and the distance between the quarterback and the receiver:
horizontal velocity (Vx) = initial velocity (V₀) * cos(angle)
Vx = 15.4 m/s * cos(38.7°)
Next, we can use the equation for horizontal motion to find the time (t) it takes for the ball to travel the horizontal distance:
distance = horizontal velocity * time
17.7 m = Vx * t
Now, let's find the vertical component of the velocity:
vertical velocity (Vy) = initial velocity (V₀) * sin(angle)
Vy = 15.4 m/s * sin(38.7°)
Since the receiver needs to catch the ball at the same height at which it was thrown, we can use the equation for vertical motion to find the time (t) it takes for the ball to reach its peak and come back down:
0 = Vy - g * t
where g is the acceleration due to gravity.
Now, we can solve for t in the second equation and substitute it into the first equation to find the horizontal distance (d):
d = Vx * t
= (15.4 m/s * cos(38.7°)) * [(2 * Vy) / g]
Finally, we can find the constant speed at which the receiver should run by dividing the horizontal distance (d) by the time (t):
constant speed = d / t
By following these steps and plugging in the given values, we can calculate the answer in units of m/s.
To find the constant speed at which the receiver should run to catch the football at the level it was thrown, we can use the following steps:
Step 1: Break down the initial velocity of the football into its horizontal and vertical components.
- The horizontal component, Vx, can be found using the equation Vx = V * cosθ, where V is the initial speed of the football and θ is the angle above the horizontal.
Vx = 15.4 m/s * cos(38.7°)
Vx = 15.4 m/s * 0.788
Vx ≈ 12.12632 m/s
- The vertical component, Vy, can be found using the equation Vy = V * sinθ, where V is the initial speed of the football and θ is the angle above the horizontal.
Vy = 15.4 m/s * sin(38.7°)
Vy = 15.4 m/s * 0.615
Vy ≈ 9.461 m/s
Step 2: Find the time it takes for the football to reach the receiver.
- Using the horizontal motion equation, d = Vx * t, and the given horizontal distance (17.7 m), we can solve for time (t).
t = d / Vx
t = 17.7 m / 12.12632 m/s
t ≈ 1.4593 s
Step 3: Determine the vertical position of the receiver when the football reaches them.
- Using the vertical motion equation, h = Vy * t + (1/2) * g * t^2, where h is the vertical position, Vy is the vertical component of initial velocity, t is time, and g is the acceleration due to gravity.
h = (9.461 m/s * 1.4593 s) + (1/2) * (9.81 m/s^2) * (1.4593 s)^2
h ≈ 13.784 m
Step 4: Calculate the constant speed at which the receiver should run.
- Since the receiver starts from a horizontal distance of 17.7 m and ends at a horizontal distance of 0 (catching the ball at the same level it was thrown), the receiver needs to cover this distance in the time it takes for the ball to reach them.
receiver's constant speed = (17.7 m) / (1.4593 s)
receiver's constant speed ≈ 12.1309 m/s
Therefore, the receiver should run at a constant speed of approximately 12.1309 m/s to catch the football at the level it was thrown.