ENERGY AND MOMENTUM: What is the final velocity of the hovering disk?

Q: A disk of mass 0.5 kg slides with a constant velocity of 2.4 m/s over an air table before colliding with an elastic band. If the band exerts an average force of 1.4 Newtons on the disk for 1.5 seconds, what is the final velocity of the disc?

A₁: The disc will experience a change in momentum ( Δp ) as a consequence of colliding with the elastic band. The change in momentum is proportional to the length of time ( t ) that the force opposes the motion of the disc ( – F = – Δp/Δt ):

FΔt = Δp

( – 1.4 N )( 1.5 s ) = – 2.1 kg*m/s

We must use the initial momentum ( p_i ) of the disc in order to determine its final momentum:

 Δp = p_f – p_i 

– 2.1 kg*m/s = p_f – ( 0.5 kg )( 2.4 m/s )

– 2.1 kg*m/s = p_f – 1.2 kg*m/s

p_f = – 0.9 kg*m/s

The negative sign indicates that a change in direction has occurred. The final velocity may now be determined:

p = mv

– 0.9 kg*m/s = ( 0.5 kg )( v_f )

v_f = – 1.8 m/s

A₂: A more analytic approach can be used with the following equation:

v_f = v_i + at

v_f = 2.4 m/s + at

F = ma

F/m = a

a = ( – 1.4 N / 0.5 kg )

a = – 2.8 m/s² 

v_f = 2.4 m/s – ( 2.8 m/s²  )( 1.5 s )

v_f = – 1.8 m/s

Q: How much time would the force need to interact with the disc to bring it to a halt?

A: v_f = v_i + at

0.0 m/s = 2.4 m/s – ( 2.8 m/s² )( t )

– 2.4 m/s = ( – 2.8 m/s² )( t )

t = 0.86 s

Check: – ( 2.8 m/s² )( 0.86 s ) = – 2.4 m/s

Additionally, Ft = Δp

Δp = ( – 1.4 N )( 0.86 s ) = – 1.2 kg*m/s

p_i = ( 0.5 kg )( 2.4 m/s ) = 1.2 kg*m/s

Δp = p_f – p_i 

– 1.2 kg*m/s = p_f – 1.2 kg*m/s

p_f = 0