A mass is given an initial velocity up a frictionless 28 deg. incline and reaches a vertical hight of 1.35m. What was the initial velocity?

To find the initial velocity of the mass, you will need to use the principles of physics, specifically the concepts of projectile motion and the conservation of energy.

The first step is to break down the given information. We are told that the incline is frictionless and inclined at an angle of 28 degrees. The mass moves up the incline and reaches a vertical height of 1.35 meters. We need to find the initial velocity.

Let's assume that the positive direction of motion is up the incline. We can then break the initial velocity into its components: one parallel to the incline (Vx) and one perpendicular to the incline (Vy).

Now, let's analyze the motion of the mass. Initially, it has both kinetic energy (due to its initial velocity) and gravitational potential energy (due to its initial vertical height). As it reaches the highest point, it will come to rest. Based on the principle of conservation of energy, we can equate the initial kinetic energy with the final potential energy.

The total initial energy (Ei) of the system is the sum of the kinetic energy (KE) and potential energy (PE):

Ei = KE + PE

The kinetic energy at the starting point can be expressed as:

KE = 1/2 * (mass) * (Vx^2 + Vy^2)

The potential energy at the highest point can be expressed as:

PE = (mass) * g * H

Where mass is the mass of the object, Vy is the vertical component of the initial velocity, Vx is the horizontal component of the initial velocity, H is the vertical height reached, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since the incline is frictionless, there is no loss of mechanical energy due to friction.

Setting the initial energy equal to the final energy (which is purely potential energy):

Ei = PE

1/2 * (mass) * (Vx^2 + Vy^2) = (mass) * g * H

Now, we need to solve this equation for the initial velocity components Vx and Vy by isolating them. Rearranging the terms, we get:

(Vx^2 + Vy^2) = 2 * g * H

Since the mass cancels out on both sides of the equation, we can eliminate mass in our calculations.

To find the initial velocity (V), we will use the Pythagorean theorem since V is the hypotenuse of a right triangle formed by Vx and Vy:

V^2 = Vx^2 + Vy^2

Squaring both sides of the equation:

V^2 = 2 * g * H

Now, we can solve for V by taking the square root of both sides:

V = sqrt(2 * g * H)

Plugging in the given values for the angle (28 degrees), acceleration due to gravity (9.8 m/s^2), and vertical height (1.35 m) into the formula, we can calculate the initial velocity.