Each of the following situations, A-D, shows a system of forces acting on a particle of given mass and the initial velocity of the particle. Work out the single extra constant force that can be added to this system, so that the particle would be brought instantaneously to rest at the specified time.

You should assume that for each situation the forces are coplanar and the motion and forces are in the same plane. You should also assume that the particle is in equilibrium in any other direction. For example, you could think of the diagrams as showing horizontal forces and motion on a horizontal surface, so the weight and normal reaction force are not shown. Alternatively, the first example could be a projectile, so we’d have motion and forces in a vertical plane.

Mass and forces Initial velocity Time when at rest
A force of 60 Newtons acting on a 6 kg particle
velocity of 8 metres per second, perpendicular to the 60 Newton force
  • If a particle is moving with a certain velocity, what do you need to do to bring it to (instantaneous) rest?

  • What if you want to bring it to rest at a specified time?

Describe how the particle behaves over time if this extra force is applied, assuming all the given forces acting on the particle remain unchanged. Compare this with how the particle would have behaved if the extra force had not been applied.

Now try these

Mass and forces Initial velocity Time when at rest
A force of 30 Newtons acting on a 5 kg particle
Velocity of 6 metres per second, 120 degrees to the 30 Newton force
A force of 12 Newtons acting on a 3kg particle to the right and a force of 14 Newtons acting to the left at 60 degrees to the line of action of the 12 Newton force
velocity of 5 metres per second, acting in the opposite direction to the 12 Newton force
Three forces acting on a 4kg particle
velocity of 8 metres per second