Standard +0.3 This is a standard two-body collision problem requiring conservation of momentum and the given speed constraint to find final velocities, then calculating kinetic energy loss. It involves straightforward algebraic manipulation with no conceptual surprises, making it slightly easier than average for A-level mechanics.
Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed 5 m s\(^{-1}\) and \(B\) is moving towards \(A\) with speed 3 m s\(^{-1}\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is 2 m s\(^{-1}\).
Find the loss of kinetic energy of the system due to the collision. [5]
Use v =v +2 or v =v −2 with their momentum equation and solve for v or v
Answer
Marks
Guidance
B A A B A B
DM1
Allow v =v 2 or v =v 2.
B A A B
v = 2.5 or v = 4.5
Answer
Marks
A B
A1
Attempt at initial KE, or final KE, or change in KE for A, or change in KE for B
1 1
Initial KE = 652 + 2(−3 )2=84
2 2
1 1
Final KE = 6(their2.5 )2 + 2(their4.5 )2
2 2
1 1
Change in KE for A= 652 − 6(their2.5 )2
2 2
1 1
Change in KE for B= 2(−3 )2− 2(their4.5 )2
Answer
Marks
Guidance
2 2
M1
Allow use or their v and/or v .
A B
Allow if 2 KE equations seen.
Answer
Marks
Guidance
Loss of KE = 45J
A1
Allow –45J.
Allow if mgv used in momentum equation.
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 2:
2 | Use conservation of momentum
65+2(−3 )=6v +2v
A B | *M1 | 4 dimensionally correct terms.
Allow sign errors, v and v must be
A B
different.
Use v =v +2 or v =v −2 with their momentum equation and solve for v or v
B A A B A B | DM1 | Allow v =v 2 or v =v 2.
B A A B
v = 2.5 or v = 4.5
A B | A1
Attempt at initial KE, or final KE, or change in KE for A, or change in KE for B
1 1
Initial KE = 652 + 2(−3 )2=84
2 2
1 1
Final KE = 6(their2.5 )2 + 2(their4.5 )2
2 2
1 1
Change in KE for A= 652 − 6(their2.5 )2
2 2
1 1
Change in KE for B= 2(−3 )2− 2(their4.5 )2
2 2 | M1 | Allow use or their v and/or v .
A B
Allow if 2 KE equations seen.
Loss of KE = 45J | A1 | Allow –45J.
Allow if mgv used in momentum equation.
5
Question | Answer | Marks | Guidance
Small smooth spheres $A$ and $B$, of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially $A$ is moving towards $B$ with speed 5 m s$^{-1}$ and $B$ is moving towards $A$ with speed 3 m s$^{-1}$. After the spheres collide, both $A$ and $B$ move in the same direction and the difference in the speeds of the spheres is 2 m s$^{-1}$.
Find the loss of kinetic energy of the system due to the collision. [5]
\hfill \mbox{\textit{CAIE M1 2022 Q2 [5]}}