Question

An object is projected vertically upward from ground level. It rises to a maximum height H. If air resistance is negligible, which of the following must be true for the object when it is at a height H/2?1. Its speed is half its initial speed 2. Its Kinetic energy is half of its initial KE 3. Its potential energy is half of its initial potential energy 4. Itrs total mechanical energy is half of its initial value 5. Its total mechanical energy is half of its value at the highest point

Answer 1

2. 2. Its Kinetic energy is half of its initial KE

Step-by-step explanation:

Kinetic and Potential Energy

The kinetic energy (K) it the capacity that an object of mass m has to do work by its speed v. It can be expressed like

`K=(mv^2)/(2)`

The potential energy (U) is the capacity of an object of mass m to do work due to its height h. The formula is

`U=mgh`

We can see the kinetic energy is not proportional to the speed, but to the square of the speed. On the other side, the potential energy is proportional to the height. Both facts will have key importance in the explanation of the answer

The problem states that an object is launched vertically upward from ground level and that it rises to a maximum height H. Let's recall an object launched upward has the maximum speed when it starts going up and loses speed until it stops in mid-air, where the speed is zero. The same thing happens to the kinetic energy, but not at the same proportion. It means that for H/2, the speed is not half the initial speed.

The potential energy is proportional to the height, so when it has just launched, U=0, but when it reaches the maximum height H, it's maximum too. So, when the height is H/2, the potential energy is also U/2.

Let's recall also that mechanical energy is conserved during the event. When h=H/2, the potential energy is half the maximum. It leaves half the mechanical energy to be all kinetic energy, so when h=H/2:

`K'=K/2,\ U'=U/2`

Let's analyze the options:

1. Its speed is half its initial speed. We have already proven it's not true since the speed is not proportional to the kinetic energy

2. Its Kinetic energy is half of its initial KE. It's true because, in the midpoint H/2, half of the maximum kinetic energy and half of the maximum potential energy are present

3. Its potential energy is half of its initial potential energy. It's false because the initial potential energy is zero. It reaches its maximum at height H

4. Its total mechanical energy is half of its initial value. It's false because the total mechanical energy is conserved, cannot be lost

5. Its total mechanical energy is half of its value at the highest point. False, the total mechanical energy is conserved, cannot be lost

The only correct option is

2. 2. Its Kinetic energy is half of its initial KE