Question

A baseball is hit from a height of 4ft at a 60∘ angle above the horizontal. Its initial velocity is 76ft per sec. (a) Write parametric equations that model the flight of the baseball.

Answer 1

To model the baseball's flight, parametric equations must account for the initial height and velocity's horizontal and vertical components. The equations as functions of time are x(t) = 38t and y(t) = -16t^2 + 66t + 4, taking gravity into consideration.

Step-by-step explanation:

To model the flight of the baseball, we must consider the initial velocity and the angle of launch. Projectile motion can be described with a set of parametric equations, and for this case, we should also factor in the initial height.

Given that the initial speed (v_0) is 76 feet per second and the angle above horizontal (θ) is 60°, we can calculate the initial horizontal (v_{0x}) and vertical (v_{0y}) components of velocity using trigonometric functions:

• v_{0x} = v_0 × cos(θ) = 76 × cos(60°)

• v_{0y} = v_0 × sin(θ) = 76 × sin(60°)

The parametric equations are then:

• x(t) = v_{0x} × t
• y(t) = -16t^2 + v_{0y} × t + 4

Here, x(t) represents the horizontal position as a function of time, and y(t) is the vertical position as a function of time. The term -16t^2 accounts for the acceleration due to gravity, which is -32 ft/s² in the vertical direction (the negative sign indicates a downward acceleration).

To input actual values:

• x(t) = 38t (since cos(60°) = 0.5)

• y(t) = -16t^2 + 66t + 4 (since sin(60°) ≈ 0.866)

The answers should be presented in feet and seconds because these are the units given in the question.