A footballl is thrown directly toward a receiver with initial speed of 18.0 m/s At an angle of 35.0 degrees above the horizontal. At that instant, The receiver is 18.0 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?
See previous post.
5m/second
To find out in what direction and with what constant speed the receiver should run to catch the football at the level at which it was thrown, we can break down the problem into two components: the horizontal and vertical motion of the football.
First, let's analyze the vertical motion of the football. We can use the equations of motion to find the time it takes for the football to reach the receiver's position.
The vertical motion of the football can be described using the following equation:
y = y0 + v0y * t - (1/2) * g * t^2
Where:
- y is the vertical displacement from the initial position (which is 0 in this case since the receiver is at the same level as the quarterback's throw).
- y0 is the initial vertical position (0).
- v0y is the initial vertical component of velocity, which can be calculated as v0 * sin(θ).
- g is the acceleration due to gravity (approximately -9.8 m/s^2, considering the downward direction).
- t is time.
The vertical displacement of the football when it reaches the receiver's position is 0. Therefore, we can rewrite the equation as follows:
0 = 0 + (v0 * sin(θ)) * t - (1/2) * g * t^2
Now, let's analyze the horizontal motion of the football. We want the horizontal displacement of the football to be equal to the distance between the quarterback and the receiver (18.0 m):
x = x0 + v0x * t
Where:
- x is the horizontal displacement.
- x0 is the initial horizontal position (which is 0 in this case since the receiver and quarterback are aligned).
- v0x is the initial horizontal component of velocity, which can be calculated as v0 * cos(θ).
- t is the same time used in the vertical motion analysis.
The horizontal displacement of the football when it reaches the receiver's position is 18.0 m according to the given information. Therefore, we can rewrite the equation as follows:
18.0 = 0 + (v0 * cos(θ)) * t
Now we have a system of two equations with two unknowns (t and v0). We can solve these equations simultaneously to find the values.
However, before solving, we should convert the angle from degrees to radians to use in the trigonometric functions. The conversion can be done using the following formula:
angle_radians = angle_degrees * (π/180)
In this case, θ = 35.0 degrees, so we have:
θ_radians = 35.0 * (π/180)
Now, we can substitute the values into the equations and solve them simultaneously to find the values of t and v0.