Skip the details
Solving motion problems through energy.
How fast at the bottom?
A ball is dropped from a high balcony. You want its speed just before it hits the ground. Following its speed second by second would be a long job.
Do you think there is a quicker way to find the final speed?
What this lesson covers
The idea
Keeping track of the total mechanical energy often gives an object's final speed or position directly, without working through the intermediate motion using Newton's laws.
How fast at the bottom?
A ball is dropped from a high balcony. You want its speed just before it hits the ground. Following its speed second by second would be a long job.
Do you think there is a quicker way to find the final speed?
- Yes, by keeping track of its energy
- No, we must follow the ball at each moment
- No, it cannot be found
Speed from height
Take g = 10 m s⁻² and ignore the air. Choose a height and a mass.
Energy does the work
We did not follow the ball through the fall. The potential energy at the top, mgh, becomes the kinetic energy at the bottom, (1/2)mv². So v² = 2gh. The mass cancels, so the speed did not change when the mass changed.
Keeping track of the total mechanical energy often gives an object's final speed or position directly, without working through the intermediate motion using Newton's laws.
Notes
Tracking the total mechanical energy often gives the final speed or position directly, without the intermediate motion.
Check yourself
A ball is dropped from 5 m. Take g = 10 m s⁻² and ignore the air. Using mgh = (1/2)mv², what is its speed just before landing, in m s⁻¹?
Answer: 10 m s⁻¹
v² = 2 × 10 × 5 = 100, so v = 10 m s⁻¹.
A ball is dropped from 45 m. Take g = 10 m s⁻² and ignore the air. What is its speed just before landing, in m s⁻¹?
Answer: 30 m s⁻¹
v² = 2 × 10 × 45 = 900, so v = 30 m s⁻¹.
To find the final speed of a falling object, keeping track of the total mechanical energy…
A ball and a heavier ball are dropped from the same height, ignoring the air. Their speeds just before landing are…
- is not possible without Newton's laws at every step. The book says energy can give the final speed without working through the intermediate motion.
- works for the speed but gives no help with the position. The book says energy can often give the final speed or the position directly.
- often gives it directly, without following the motion step by step — correct. Yes!
- the same — correct. Yes!
- greater for the heavier ball. In mgh = (1/2)mv² the mass cancels, so v² = 2gh depends on the height.
- greater for the lighter ball. The mass cancels, so the speed depends only on the height.