Calculating Kinetic Energy at the Bottom of an Inclined Plane Using Energy Conservation

When a mass slides from rest on a frictionless inclined plane, its gravitational potential energy is transformed into kinetic energy as it reaches the bottom of the slope. This problem can be solved by applying the principle of conservation of energy, which states that energy cannot be created or destroyed, only converted from one form to another.

Step-by-Step Solution:

Step 1: Calculate the Potential Energy (PE) at the Top of the Incline

Using the formula for potential energy, PE mgh, where:

m is the mass of the block, g is the acceleration due to gravity, and h is the height of the incline.

Given: m 2 kg, g 10 m/s^2 (using the value for Earth's gravity), and h 10 m.

Substituting the values, we get:

PE 2 kg times; 10 m/s^2 times; 10 m 200 J (Joules).

Step 2: Calculate the Kinetic Energy (KE) at the Bottom of the Incline

Since there is no friction, all of the potential energy at the top is converted into kinetic energy at the bottom. Therefore, KE PE.

KE PE 200 J.

Additional Insights and Variations:

Let's consider an alternative approach to solving this problem, acknowledging the ambiguity in the question regarding the dimensions of the incline.

Scenario 1: The Inclined Plane is 20m Long and 10m High (Diagonally)

a g times; sin(U), where U is the angle of incline relative to the horizontal. t sqrt(2s/a), where s is the distance along the slope. K.E. 1/2 × m × at^2 1/2 × m × (g times; sin(U))^2 × (2s/gtimes;sin(U)) m × g × sin(U) × s.

Given sin(U) 10 m / 20 m 0.5 for the diagonally measured distance, and knowing s 20 m along the slope, we can calculate:

K.E. 2 kg times; 9.81 m/s^2 times; 0.5 times; 20 m 196.2 kgmiddot;m^2/s^2 196.2 J.

Scenario 2: The Inclined Plane is 20m Long and 10m High (Horizontally)

s 20 m times; cos(U) 20 m times; 20 / sqrt(10^2 20^2) 40 m/sqrt(5). sin(U) 10 / sqrt(10^2 20^2) 1/sqrt(5). K.E. 2 kg times; 9.81 m/s^2 times; 1/sqrt(5) times; 40 m/sqrt(5) 156.94 J.

This problem highlights the importance of accurately interpreting given dimensions and applying the principles of physics and energy conservation appropriately.

Conclusion:

Whether we are dealing with a 20m long and 10m high slope diagonally or horizontally, the principle of conservation of energy holds. The kinetic energy at the bottom of the slope, in either case, is 196.2 J or approximately 156.94 J, respectively.

Note: Both scenarios demonstrate the importance of understanding the geometry of the system and accurately applying physical principles.