Question

Jeremy stands on the edge of a cliff. He throws three identical rocks with the same speed. Rock X is thrown vertically upward, rock Y is thrown horizontally, and rock Z is thrown vertically downward. If the ground at the base of the cliff is level, which rock hits the ground with the greatest speed if there is no air resistance?

Answer 1

All the three rocks will hit the ground with same speed.

Step-by-step explanation:

For rocks X and Z, motion is along a straight line but in case of rock Y, motion is two dimensional. Since velocity is a vector it will be difficult for us to calculate the final velocity in each case. So we should find a way to solve this problems using a scalar which is related to velocity. The best and easy to use scalar related to velocity is kinetic energy. Since there is no air resistance, the total mechanical energy of the stone remains the same. Therefore we can use the concept of conservation of mechanical energy to solve this problem.

i.e. initial mechanical energy = final mechanical energy

let us take the edge of the cliff as initial position and ground as the final position.

We know that

Mechanical energy = Kinetic energy + Potential energy

Initial Mechanical energy = Initial Kinetic energy + Initial Potential energy

we know that

Potential energy = mgh

where,

m = mass of the body

g = acceleration due to gravity

h = height from ground

All the three rocks are identical and are thrown from same height. Therefore m and h are same for all the three which implies that the initial potential energy for all the three rocks is same.

Similarly, we know that

Kinetic energy =
`(1)/(2) mv^(2)`

where,

m = mass of the body

v = velocity of the body

Since all the rocks are thrown with same speed, v is same for all the rocks. Thus initial kinetic energy is also same for all.

Since initial kinetic energy and Initial Potential energy is same for all the three, Initial Mechanical energy is also same for them.

Next let us consider the final position. At the ground h = 0. Therefore final potential energy of all the three rocks is 0. Thus they will be having only kinetic energy.

By conservation of mechanical energy,

initial mechanical energy = final mechanical energy

i.e. Initial Kinetic energy + Initial Potential energy = final Kinetic energy + final Potential energy

final potential energy = 0

thus,

Initial Mechanical energy = Initial Kinetic energy + Initial Potential energy = final Kinetic energy

Initial Mechanical energy = final Kinetic energy

Since Initial Mechanical energy is same for all the three, by the above equation final Kinetic energy is also same for all the three. Since here, kinetic energy is the function of only velocity, final velocity is also same for all the three rocks.

i.e. all the three rocks will hit the ground with same speed.