A block is projected up a frictionless inclined plane with initial speed v0 = 8.42 m/s. The angle of incline is è = 58.4°. (a) How far up the plane does it go? (b) How long does it take to get there? (c) What is its speed when it gets back to the bottom?

To solve this problem, we can use the principles of physics and trigonometry. Let's break down each part of the problem.

(a) How far up the plane does it go?
To find how far up the plane the block goes, we need to determine the displacement along the incline.

The initial velocity along the incline (V₀x) is given by: V₀x = V₀ * cos(θ), where V₀ is the initial speed and θ is the angle of incline.

Substituting the given values: V₀x = 8.42 m/s * cos(58.4°)

Now, we can calculate the displacement along the incline (s) using the formula: s = V₀x^2 / (2 * a), where a is the acceleration along the incline.

Since the inclined plane is frictionless, the only force acting on the block along the incline is the component of gravity parallel to the incline. This force is given by: F = m * g * sin(θ), where m is the mass of the block and g is the acceleration due to gravity.

The acceleration along the incline (a) is given by: a = F / m = g * sin(θ).

Substituting the values, we have: s = (V₀x^2) / (2 * g * sin(θ))

(b) How long does it take to get there?
To find the time it takes for the block to reach the highest point of the incline, we can use the formula for time (t) in motion on inclines:

t = V₀x / a.

Substituting the values, we have: t = V₀x / (g * sin(θ))

(c) What is its speed when it gets back to the bottom?
To find the speed of the block when it gets back to the bottom, we can use the conservation of mechanical energy. Since there is no friction, the mechanical energy is conserved.

The potential energy at the highest point is given by: PE = m * g * h, where h is the height above the bottommost point of the incline.

The initial kinetic energy is given by: KE₀ = (1/2) * m * V₀²

The final kinetic energy at the bottom is given by: KE = (1/2) * m * V²

Since the total mechanical energy is conserved, we can equate the potential energy at the highest point to the kinetic energy at the bottom:

PE = KE
m * g * h = (1/2) * m * V²

Simplifying, we find: V = √(2 * g * h)

Substituting the values for g (acceleration due to gravity) and h (height), we can calculate the speed when the block gets back to the bottom.

To solve this problem, we will use the equations of motion for projectile motion on an inclined plane.

(a) To find how far up the plane the block goes, we can use the equation:

s = (v0^2 * sin(2è)) / g

where s is the distance traveled up the plane, v0 is the initial speed, è is the angle of incline, and g is the acceleration due to gravity.

Plugging in the given values:

s = (8.42^2 * sin(2 * 58.4°)) / 9.8

s ≈ 13.8 meters

Therefore, the block goes approximately 13.8 meters up the plane.

(b) To find the time taken to reach that point, we can use the equation:

t = (v0 * sin(è)) / g

where t is the time taken, v0 is the initial speed, è is the angle of incline, and g is the acceleration due to gravity.

Plugging in the given values:

t = (8.42 * sin(58.4°)) / 9.8

t ≈ 0.815 seconds

Therefore, it takes approximately 0.815 seconds to reach that point.

(c) To find the speed when it gets back to the bottom, we can use the equation:

v = v0 * cos(è)

where v is the final speed, v0 is the initial speed, and è is the angle of incline.

Plugging in the given values:

v = 8.42 * cos(58.4°)

v ≈ 4.29 m/s

Therefore, the speed of the block when it gets back to the bottom is approximately 4.29 m/s.