Question

A quarterback throws a football with an initial velocity v at an angle θ above horizontal. Assume the ball leaves the quarterback’s hand at ground level and moves without air resistance. All portions of this problem will produce algebraic expressions in terms of v, θ, and g. Let the origin of the Cartesian coordinate system be the ballʼs initial position.

(a): Write an expression for the magnitude of the football’s initial vertical velocity v0y.
(b): Find an expression for the magnitude of the football’s initial horizontal velocity v0x.

Answer 1

The initial vertical velocity of the football is v*sin(θ) and the initial horizontal velocity is v*cos(θ). These components are derived using trigonometric functions, under the assumption of no air resistance.

Step-by-step explanation:

To answer the student's question, we must consider the components of the initial velocity of the football thrown by the quarterback. The initial velocity v at angle θ above the horizontal can be resolved into vertical and horizontal components using basic trigonometry:

(a) The magnitude of the initial vertical velocity can be expressed with the sine function:

v0y = v * sin(θ)

(b) The magnitude of the initial horizontal velocity can be expressed with the cosine function:

v0x = v * cos(θ)

These expressions are derived assuming the absence of air resistance, and g represents the acceleration due to gravity.

Answer 2

(a) The y-component or vertical velocity is calculated using:
Vy = Vsin(∅)

(b) The x-component or horizontal velocity is calculated using:
Vx = Vcos(∅)