Question

A lamp hangs from the ceiling at a height of 2.9 m. If the lamp breaks and falls to the floor, what is its impact speed?

Answer 1

The impact speed of the lamp is `~~7.54m/s`.

Explanation:

You can do this calculation using at least two different methods: kinematics and energy conservation.

Kinematics

The lamp experiences free-fall for a brief period of time when it falls, so its vertical acceleration is equal to `-g`, where `g` is the free-fall (gravitational) acceleration, `9.8m/s`.

We are given that the initial position of the lamp is `y_i=2.9m` vertically, and when it hits the floor, it is at a final position of `y_f=0m`.

Additionally, the lamp is initially at rest when it hangs from the ceiling, so `v_i=0m/s`.

We want to find `v_f`. Assuming no air resistance, we can calculate this value using a kinematic equation. Specifically:

`(v_(fy))^2=(v_(iy))^2+2a_yDeltay`

`=>(v_(fy))^2=cancel((v_(iy))^2)+2a_yDeltay`

Solving for `v_(fy)`:

`v_(fy)=sqrt(2a_yDeltay)`

Using our known values:

`v_(fy)=sqrt(2(-9.8m/s^2)(0m-2.9m))`

`v_(fy)~~7.54m/s`

The impact speed of the lamp is therefore `~~7.54m/s`. The impact velocity would be `-7.54m/s` as the lamp is moving in the negative direction, i.e. downward.

Energy Conservation

Initially, the lamp has only gravitational potential energy.

`U_g=mgh`

Just before it hits the ground, virtually all of this potential energy has been transformed into kinetic energy:

`K=1/2mv^2`

If we say that energy is conserved (in reality it is not), the potential energy stored within the lamp initially is equal to the kinetic energy stored in the lamp just before it hits the ground.

`U_g=K`

`cancelmgh=1/2cancelmv^2`

Solving for `v`:

`v=sqrt(2gh)`

`v=sqrt(2*9.8m/s^2*2.9m)`

`v~~7.54m/s`

Once again, if the problem asked for the impact velocity, you would include a negative sign to show that the object is traveling in the negative direction, i.e. downwards.

As mentioned above, energy is not conserved in reality. Air resistance (drag) plays a role, even if it is minimal, and energy is also lost to factors such as the sound produced as the lamp (assumably) breaks. These are approximations.