What minimum speed does a 160 puck need to make it to the top of a 5.4 -long, 24 frictionless ramp?

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To determine the minimum speed required for a puck to make it to the top of a ramp, we need to consider the forces acting on the puck.

First, we'll assume there is no air resistance and the ramp is frictionless. This means the only force acting on the puck is its weight, which is given by the equation:

Weight (W) = mass (m) * acceleration due to gravity (g)

Since the puck is on a ramp, we'll resolve the weight force into two components: one parallel to the ramp and one perpendicular to the ramp. The component parallel to the ramp will be responsible for the puck's motion up the ramp, while the perpendicular component will be balanced by the normal force from the ramp.

The component of the weight force parallel to the ramp is given by:

Weight parallel to the ramp = Weight * sin(theta)

Where theta is the angle of the ramp.

Now, we know that the work done by the parallel force to move the puck up the ramp is equal to the change in the puck's potential energy. The work done can be calculated using the following equation:

Work = force * distance

The force acting parallel to the ramp is the weight parallel to the ramp. The distance is the length of the ramp, which is given as 5.4 m in this case. Therefore:

Work = (Weight parallel to the ramp) * distance

Since the change in potential energy is equal to the work done, we can write:

Change in potential energy = (Weight parallel to the ramp) * distance

Now, we'll equate this to the change in kinetic energy for the puck to reach the top of the ramp:

Change in potential energy = Change in kinetic energy

For the puck to reach the top, its final kinetic energy at the top should be zero, as it comes to a stop. Therefore, we can write:

Change in potential energy = Initial kinetic energy

The initial kinetic energy is given by the equation:

Initial kinetic energy = (1/2) * mass * velocity^2

So, we have:

(Weight parallel to the ramp) * distance = (1/2) * mass * velocity^2

Plugging in the given values: distance = 5.4 m, angle of the ramp theta = 24 degrees, and acceleration due to gravity g = 9.8 m/s^2, we can rearrange the equation to solve for the minimum velocity (velocity required for the puck to reach the top):

Velocity = sqrt((2 * Weight parallel to the ramp * distance) / mass)

We'll need the mass of the puck to calculate the minimum velocity. Once we have that, we can substitute the values into the equation to find the answer.