An arrow is shot into the air and moves along the parabolic path y=x(50-x). The horizontal component of velocity is always 90 ft/s. What is the vertical component of velocity when...

x=10
x=40

Why did the arrow go to a comedy show? Because it wanted to find its funny "x"-factor!

When x = 10, let's calculate the vertical component of velocity. Using the equation y = x(50 - x), we have y = 10(50 - 10). Simplifying, we get y = 10(40) = 400 ft. So, the arrow is at a height of 400 ft.

Since we know the horizontal component of velocity is always 90 ft/s and there is no horizontal acceleration, we can say the vertical component of velocity remains constant throughout the motion.

Therefore, the vertical component of velocity when x = 10 is the same as the initial vertical component of velocity, which we don't have enough information to determine. It's a bit of a cliffhanger!

Now let's move on to x = 40. Inserting x = 40 into the equation y = x(50 - x), we have y = 40(50 - 40). Simplifying, we get y = 40(10) = 400 ft. So, once again, the arrow is at a height of 400 ft.

Similarly, since there is no horizontal acceleration, the vertical component of velocity when x = 40 remains the same as the initial vertical component of velocity, which, again, we don't have enough information to calculate. It's like a real-life suspense movie!

Remember, this is all assuming ideal conditions and ignoring factors like air resistance and wind. Don't shoot the messenger!

To find the vertical component of velocity, we need to find the derivative of the equation y = x(50-x) with respect to time.

The given equation represents the height (y) of the arrow at different positions (x).

Let's find the derivative of y with respect to x, and then we can isolate the vertical component of velocity:

dy/dx = d/dx(x(50-x))
= d/dx(50x - x^2)
= 50 - 2x

Now, the vertical component of velocity (Vy) is given by:

Vy = dy/dt

To find the vertical component of velocity when x = 10, substitute x = 10 into the equation Vy = 50 - 2x:

Vy = 50 - 2(10)
Vy = 50 - 20
Vy = 30 ft/s

Therefore, when x = 10, the vertical component of velocity is 30 ft/s.

To find the vertical component of velocity when x = 40, substitute x = 40 into the equation Vy = 50 - 2x:

Vy = 50 - 2(40)
Vy = 50 - 80
Vy = -30 ft/s

Therefore, when x = 40, the vertical component of velocity is -30 ft/s.

To find the vertical component of velocity at a given value of x, we need to first differentiate the equation y = x(50 - x) with respect to x in order to find the equation for the vertical velocity.

Let's differentiate y = x(50 - x) using the product rule of differentiation:

d/dx (y) = d/dx (x(50 - x))
= 50x - x^2

Now we have the equation for the vertical velocity, which is the time derivative of the vertical position. In physics, the vertical velocity is commonly represented as vy.

vy = 50x - x^2

To find the vertical component of velocity when x = 10, we substitute the value of x into the equation:

vy = 50(10) - (10)^2
= 500 - 100
= 400 ft/s

Therefore, the vertical component of velocity when x = 10 is 400 ft/s.

To find the vertical component of velocity when x = 40, we again substitute the value of x into the equation:

vy = 50(40) - (40)^2
= 2000 - 1600
= 400 ft/s

Therefore, the vertical component of velocity when x = 40 is also 400 ft/s.

find dy/dx to find the slope (that is the velocity)

you know that dx=90