Parabolic ​motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>

Verified step by step guidance

1. First, understand that the path of the arrow is given by the equation y = x(50 - x). This is a parabolic equation representing the vertical position y as a function of the horizontal position x.
2. To find the vertical component of velocity, we need to determine the derivative of y with respect to time t, which is dy/dt. However, y is given as a function of x, so we will use the chain rule: dy/dt = (dy/dx) * (dx/dt).
3. The horizontal component of velocity is given as dx/dt = 30 ft/s. This is constant throughout the motion.
4. Next, find dy/dx by differentiating y = x(50 - x) with respect to x. This gives dy/dx = 50 - 2x.
5. Now, substitute dy/dx and dx/dt into the chain rule expression: dy/dt = (50 - 2x) * 30. This expression gives the vertical component of velocity as a function of x. Evaluate this expression for (a) x = 10 and (b) x = 40 to find the vertical component of velocity at these points.

Key Concepts

Parabolic Motion

Parabolic motion refers to the trajectory of an object that is influenced by gravity, resulting in a curved path. In this case, the motion of the arrow follows a parabolic equation, which describes how the vertical position (y) changes with respect to the horizontal position (x). Understanding this concept is crucial for analyzing the motion of projectiles and determining their velocity components.

Velocity Components

Velocity is a vector quantity that has both magnitude and direction. In projectile motion, the total velocity can be broken down into horizontal and vertical components. The horizontal component remains constant (30 ft/s in this case), while the vertical component changes due to gravitational acceleration. Calculating these components at specific points along the trajectory is essential for understanding the arrow's motion.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. In the context of this problem, differentiating the parabolic equation y = x(50 - x) with respect to x allows us to determine the vertical component of velocity, which is the derivative of the height with respect to time. This mathematical tool is vital for analyzing motion and finding instantaneous rates.