Question

In which situation is mechanical energy conserved?

A. a parachutist descends at constant speed

B. a car slows down without braking along a flat road

C. a heavy crate is raised at constant speed via a rope and pulley system

D. a sled accelerates as it slides down a frictionless inclined plane

____________________

Two spaceships have the same mass. Spaceship A has kinetic energy K and spaceship B has kinetic energy 16K. What is the ratio of their speeds, vA/vB?

A. 1/16

B. 1/8

C. 2/1

D.  1/4

Answer 1

Note:

  

  

And The total mechanical energy = Kinetic energy + Potential energy

Where, m is mass of the object

g = 9.8 m/s² is acceleration due to gravity

h is height of the object from a reference level (generally surface of Earth)

V is speed of the object

Question 1:

(A) If the parachutist descends at constant speed then, the kinetic energy will not change (Because speed is constant) but the potential energy will change (because height of the parachutist is changing). Hence the total mechanical energy will not remain conserved.

(B) If a car is slowing down without braking on a flat road, it means that the speed of the car is decreasing i.e. the kinetic energy of the car is decreasing but potential energy of the car will not change (as height is not changing). Hence, the total mechanical energy will not remain cconserved.

(C) Again the crate is raised by the pulley with constant speed , i.e. the kinetic energy will not change (as speed is constant) but the potential energy will change (because height of the crate is changing ). Hence, the total mechanical energy will not remain conserved.

(D) A sled accelerates as it slides down a frictionless inclined plane :

In this case the kinetic energy is changing (because sled is accelerating , it means that speed of sled is changing so the kinetic energy) at the cost of potential energy. Because the acceleration in the sled is only due to the change of height of sled.

And if a body is accelerating, it means that the kinetic energy of the body is increasing. And if a body is sliding down, it means height of the body is decreasing (from a reference level below) i.e. potential energy is decreasing.

it means that, in the given situation the kinetic energy of the sled is increasing but the potential energy is decreasing with time.

So, if we calculate total mechanical energy of the system at any two point, it will never change. Hence, this is the only case in which the total mechanical energy is conserved.

Question 2:

Given,

The two spaceship has same mass i.e.

mass of A (m_A) = mass of B (m_B) = m (say) ,

Kinetic energy of spaceship A = K ,

Kinetic energy of spaceship B = 16 K ,

Speed of spaceship A = V_A

and Speed of spaceship B = V_B

So, using equation (1), we can write that   

....................................(2)

and ....................................(3)

dividing equation(2) by equation (3), we get

  

or,

or,

So, the correct option is 'D'.

For any doubt please comment and please give an up vote. Thank you.