03 2-1 (c) Acceleration as a function of position
Method
- If the force depends on the distance from a fixed point, use .
- Separate: , i.e. .
- To find the time afterwards, use and integrate .
✍️ Worked examples
Example 6:
- A body of mass 2 kg starts from rest at the origin under the force N along the -axis ( in metres).
- Find in terms of , the greatest speed, and where the body first stops.
Solution:
- , so .
- gives , so .
- The speed is greatest where , i.e. m: , so m/s.
- The body stops when : , so m. There the force is N, so it turns back.
Example 7:
- A body of mass 1 kg is released from rest at the origin and acted on by N, with in metres.
- Find when m, and when m/s.
Solution:
- , so (the constant is 0 since at ).
- At : , so and m/s.
- At : , so and .
- m (the root is rejected).
Example 8:
- A particle of mass 1 kg moves along a line under an attractive force towards the origin of magnitude N at distance .
- At its speed is 6 m/s.
- Show that it stops at m.
Solution:
- The force is towards the origin, so : .
- , so .
- when , i.e. m.
- The motion is an oscillation between and (simple harmonic with period s).
Example 9: A body falls from rest with acceleration where is the distance fallen. (Take and m.) Find its speed when m.
Solution:
- , so .
- At : , so .
- , so m/s.