Momentum: Mathematics 9709 (Cambridge International AS & A Level)

Two toy cars, $P$ and $Q$, each of mass $0.5\text{ kg}$, travel towards each other along the same straight, smooth, horizontal track. Immediately before impact, $P$ has velocity $3\text{ m s}^{-1}$ and $Q$ has velocity $3\text{ m s}^{-1}$ in the opposite direction. On collision, $P$ and $Q$ lock together and move as a single combined toy.

Taking the direction of $P$'s initial motion as positive, what is the common velocity of $P$ and $Q$ immediately after the collision?

Two smooth spheres, $A$ and $B$, move in the same straight line on a smooth horizontal table. Sphere $A$ has mass $3\text{ kg}$ and moves at $8\text{ m s}^{-1}$. Sphere $B$ has mass $5\text{ kg}$ and moves at $2\text{ m s}^{-1}$ in the same direction, some distance ahead of $A$. $A$ catches up with $B$ and they collide directly.

(a) Taking the common direction of motion as positive, write down the total momentum of the system before the collision. [1]

(b) Immediately after the collision, $A$ has velocity $1\text{ m s}^{-1}$ in this same positive direction. Find the velocity of $B$ immediately after the collision, stating clearly the direction in which $B$ moves. [3]

Railway truck $P$ has mass $800\text{ kg}$ and moves at $5\text{ m s}^{-1}$ along a straight horizontal track. Truck $Q$ has mass $1200\text{ kg}$ and moves at $2\text{ m s}^{-1}$ along the same track, directly towards $P$. The trucks collide and couple together automatically, moving as one combined truck immediately after impact.

(a) Taking the direction of $P$'s initial motion as positive, write down the signed initial velocities of $P$ and $Q$, and hence state the total momentum of the system before the collision. [2]

(b) Find the common velocity of the coupled trucks immediately after the collision, stating the direction in which they move. [3]

(c) State, with a reason but without further calculation, whether the coupled trucks would move faster or slower immediately after collision if $Q$ had instead been moving at $2\text{ m s}^{-1}$ in the same direction as $P$ (rather than towards $P$). [1]

Two ice skaters, $A$ of mass $55\text{ kg}$ and $B$ of mass $70\text{ kg}$, stand at rest facing each other on frictionless ice. They push off from each other, moving apart along the same straight line.

(a) Taking the direction in which $A$ moves off as positive, write down the total momentum of the system immediately before they push apart. [1]

(b) Given that $A$ moves off at $3.5\text{ m s}^{-1}$, find the velocity of $B$ immediately after they push apart, stating clearly the direction in which $B$ moves. [3]

(c) Explain, using conservation of momentum, why $A$ and $B$ must move off in opposite directions. [1]

Three gliders, $A$, $B$ and $C$, move on a straight, smooth, horizontal air track. Glider $A$ has mass $0.6\text{ kg}$ and moves at $4\text{ m s}^{-1}$ towards glider $B$, which has mass $0.9\text{ kg}$ and is initially at rest. $A$ and $B$ collide directly; they do not coalesce.

(a) Given that immediately after this first collision $A$ has velocity $-0.2\text{ m s}^{-1}$ (that is, it rebounds), find the velocity of $B$ immediately after the collision. [3]

(b) Glider $B$, now moving with the velocity found in part (a), goes on to collide with glider $C$, mass $1.5\text{ kg}$, which is at rest further along the track. $B$ and $C$ coalesce on impact. Find the common velocity of $B$ and $C$ immediately after this second collision. [3]

(c) State the direction in which $A$ moves after its collision with $B$, and explain whether $A$ can ever catch up with the combined glider formed by $B$ and $C$. [2]

A delivery drone of mass $4\text{ kg}$ flies horizontally in a straight line at a constant velocity of $12\text{ m s}^{-1}$, directly away from its depot.

What is the magnitude of the drone's momentum?

Two go-karts, $A$ and $B$, move in the same straight line on a smooth horizontal track. Go-kart $A$ has mass $180\text{ kg}$ (including its driver) and moves at $6\text{ m s}^{-1}$. Go-kart $B$ has mass $m\text{ kg}$ (including its driver) and moves at $4\text{ m s}^{-1}$ in the same direction, some distance ahead of $A$. $A$ catches up with $B$ and they collide directly, without coalescing. Immediately after the collision, $A$ has velocity $3\text{ m s}^{-1}$ and $B$ has velocity $10\text{ m s}^{-1}$, both in the same (original) direction.

(a) Taking the common direction of motion as positive, form an equation in $m$ using conservation of momentum, and hence find the mass $m$ of go-kart $B$. [4]

(b) State, with a reason, whether go-kart $A$ can collide with go-kart $B$ again after this collision. [2]

At a fairground, bumper car $C$ (mass $250\text{ kg}$ with its rider) moves at $2\text{ m s}^{-1}$ along a straight track. Bumper car $D$ (mass $300\text{ kg}$ with its rider) moves at $1.5\text{ m s}^{-1}$ directly towards $C$ along the same track. The cars collide directly and do not coalesce.

(a) Taking the direction of $C$'s initial motion as positive, write down the signed initial velocities of $C$ and $D$, and hence find the total momentum of the system before the collision. [1]

(b) Given that immediately after the collision $C$ has velocity $-1\text{ m s}^{-1}$ (that is, it rebounds), find the velocity of $D$ immediately after the collision. [4]

(c) State the direction each car moves in immediately after the collision, and explain whether the two cars can collide with each other again. [2]

Two shopping trolleys, $X$ and $Y$, roll in the same straight line across a smooth, flat, horizontal car park. Trolley $X$ has mass $15\text{ kg}$ and moves at $2\text{ m s}^{-1}$. Trolley $Y$ has mass $10\text{ kg}$ and moves at $0.5\text{ m s}^{-1}$ in the same direction, some distance ahead of $X$. $X$ catches up with $Y$ and the two trolleys collide and lock together, moving as a single combined trolley immediately after impact.

(a) Taking the common direction of motion as positive, find the total momentum of the system before the collision. [2]

(b) Find the common velocity of the combined trolley immediately after the collision. [3]

Two magnetic toy trains, $P$ and $Q$, run on the same straight track. $P$ has mass $0.8\text{ kg}$ and moves at $5\text{ m s}^{-1}$ towards $Q$, which has mass $m\text{ kg}$ and is initially at rest. $P$ and $Q$ collide and couple together, moving as a single combined train at $2\text{ m s}^{-1}$ immediately after the collision.

What is the mass $m$ of train $Q$?