A sled is initially given a shove up a frictionless 24.0° incline. It reaches a maximum vertical height 1.10 m higher than where it started. What was its initial speed?

1/2 m v^2= gaininPE=mg*1.1

solve for initial velocity, v

To find the initial speed of the sled, we can use two key principles: conservation of energy and trigonometry.

First, let's consider the conservation of energy. At the maximum height, the sled has gained potential energy (due to the increased height) and lost kinetic energy (due to a decrease in speed).

The potential energy gained can be calculated using the equation:
Potential Energy = mass * gravity * height

Since the incline is frictionless, we can assume there is no energy loss due to friction. Therefore, the potential energy gained equals the kinetic energy lost.

Next, we can express the height relative to the starting point (h = 1.1 m) in terms of the slope length (d). This can be done using trigonometry, specifically the sine function:

sin(θ) = opposite/hypotenuse
sin(24°) = h/d

Rearranging the equation gives:
d = h / sin(24°)

Now that we have the height expressed in terms of the slope length, we can substitute it into the equation for potential energy:

Potential Energy = mass * gravity * (h / sin(24°))

Since potential energy equals kinetic energy lost, we can equate them:

1/2 * mass * initial speed^2 = mass * gravity * (h / sin(24°))

Mass gets canceled out, and we can solve for the initial speed:

initial speed^2 = 2 * gravity * h / sin(24°)

Taking the square root of both sides gives:

initial speed = sqrt(2 * gravity * h / sin(24°))

Now we can plug in the known values:
gravity = 9.8 m/s^2
h = 1.1 m
θ = 24°

Calculating the equation yields the initial speed of the sled.