Question

A ball is launched with initial speed v from ground level up a frictionless slope. The slope makes an angle with the horizontal. Using conservation of energy, find the maximum vertical height hₘₐₓ to which the ball will climb.

Express your answer in terms v, g, and 0, as needed.

Answer 1

To find the maximum vertical height a ball will climb, set the initial kinetic energy equal to the final potential energy. The mass cancels out, and the maximum height depends only on the initial speed and the acceleration due to gravity.

Step-by-step explanation:

To calculate the maximum vertical height hmax to which a ball will climb when launched up a frictionless slope, we can use the principle of conservation of energy. The total mechanical energy (kinetic plus potential energy) at launch must be equal to the total mechanical energy at the ball's highest point, because no energy is lost to friction.

Initially, the ball has kinetic energy given by KEi = 0.5 * m * v2, where m is the mass of the ball and v is its launch speed. At the highest point, all the kinetic energy is converted to gravitational potential energy given by PEf = m * g * hmax, where g is the acceleration due to gravity and hmax is the maximum height.

Setting the initial kinetic energy equal to the final potential energy, we get:

• 0.5 * m * v2 = m * g * hmax

Now, we can solve for hmax:

• hmax = (0.5 * v2) / g

Notice how the mass m cancels out, and the maximum height is only dependent on the initial speed v, the acceleration due to gravity g, and the slope angle is not needed since the question asks for vertical height.