Dynamics

State the principle of conservation of momentum.

Calculate:

velocity $v$

$v$ = ______ $\text{m s}^{-1}$

the percentage of the total initial kinetic energy of the two objects that is transferred to other forms of energy during the collision.

percentage = ______ $\%$

Define momentum.

Calculate the change in momentum of the object from time $t = 0$ to $t = 12\text{ s}$.

change in momentum = ______ $\text{kg m s}^{-1}$

Calculate the magnitude of the resultant force acting on the object.

force = ______ $\text{N}$

Describe the variation of the speed of the object from time $t = 0$ to $t = 8.0\text{ s}$.

By reference to Fig. 2.1, explain why the resultant force acting on the object during the first $8.0\text{ s}$ of its motion cannot be due to air resistance.

At time $t = 0$ the displacement $d$ of the object is zero.

On Fig. 2.2, sketch the variation of $d$ with time $t$ from $t = 0$ to $t = 12\text{ s}$.

Numerical values of $d$ are not required.

State the principle of conservation of momentum.

An object of mass $2m$ is travelling at a speed of $5.0\text{ m s}^{-1}$ in a straight line. It collides with an object of mass $3m$ which is initially stationary, as shown in Fig. 3.1.

After the collision, the object of mass $2m$ moves with velocity $v$ at an angle of $30^\circ$ to its original direction of motion.

The object of mass $3m$ moves with velocity $w$ also at an angle of $30^\circ$, as shown in Fig. 3.2.

By considering the conservation of momentum in two dimensions, calculate the magnitudes of $v$ and $w$.

$v$ = ______ $\text{m s}^{-1}$
$w$ = ______ $\text{m s}^{-1}$

An object of mass $4.2\text{ kg}$ is travelling in a straight line at a speed of $6.0\text{ m s}^{-1}$. The object is brought to rest in a distance of $0.050\text{ m}$ by a constant force.

Calculate the magnitude of this force.

$\text{force}$ = ______ $\text{N}$

Define linear momentum.

Calculate the maximum speed reached by the car.

maximum speed = ______ $\text{m s}^{-1}$

Calculate the maximum kinetic energy of the car.

maximum kinetic energy = ______ $\text{J}$

Show that the acceleration of the car at time $t = 4.0\text{ s}$ is $5.0\text{ m s}^{-2}$.

Determine the distance travelled by the car between times $t = 0$ and $t = 12.0\text{ s}$.

distance = ______ $\text{m}$

On Fig. 2.2, sketch the variation with time $t$ of the acceleration $a$ of the car in (b) from $t = 0$ to $t = 12.0\text{ s}$.

State the principle of conservation of momentum.

Show that the speed $v$ of ball Y after the collision is $8.0\ \text{m s}^{-1}$.

Calculate the change in the total kinetic energy $\Delta E_K$ of the balls due to the collision.

$\Delta E_K$ = ______ $\text{J}$

Determine the magnitude and direction of the force exerted on ball X by ball Y during the collision.

magnitude = ______ $\text{N}$
direction ______

Compare the magnitude and direction of the force exerted on ball Y by ball X during the collision with the answers in (c)(i). No further calculations are required.

Block X has an initial kinetic energy of $0.30\text{ J}$.

Calculate the mass of block X.

mass = ______ $\text{kg}$

Determine the magnitude of the momentum of block Y after the collision.

momentum = ______ $\text{kg m s}^{-1}$

Block X exerts an average force of $7.7\text{ N}$ on block Y during the collision.

Calculate the time that the blocks are in contact with each other.

time = ______ $\text{s}$

State Newton’s second law of motion.

the resultant force on the block

resultant force = ______ $\text{N}$

the force $X$.

$X$ = ______ $\text{N}$

On Fig. 3.3, sketch a graph to show the variation of force $X$ with time $t$ from $t = 0$ to $t = 6.0\ \text{s}.

Define force.

Determine the change in momentum of the block from time $t = 0$ to time $t = 3.0\ \text{s}$.

change in momentum = ______ $\text{kg m s}^{-1}$

Describe and explain the motion of the block between time $t = 3.0\ \text{s}$ and time $t = 6.0\ \text{s}$.

Force $X$ produces a total power of $2.0\ \text{W}$ when moving the block between time $t = 3.0\ \text{s}$ and time $t = 6.0\ \text{s}$.

Calculate the distance moved by the block during this time interval.

distance = ______ $\text{m}$

The block is at rest at time $t = 0$.

On Fig. 3.3, sketch a graph to show the variation of the momentum of the block with time $t$ from $t = 0$ to $t = 6.0\ \text{s}$.
Numerical values of momentum are not required.

Calculate the component of the final momentum of ball Y in the direction perpendicular to line PQ.

component of momentum = ______ $\text{Ns}$

By considering the component of the final momentum of each ball in the direction perpendicular to line PQ, calculate $m_Z$.

$m_Z$ = ______ $\text{kg}$

During the collision, the average force exerted on Y by Z is $F_Y$ and the average force exerted on Z by Y is $F_Z$.

Compare the magnitudes and directions of $F_Y$ and $F_Z$. Numerical values are not required.

magnitudes: .......................................................................................................................

directions: ..........................................................................................................................

Two blocks, A and B, move directly towards each other along a horizontal frictionless surface, as shown in the view from above in Fig. 4.3.

The blocks collide perfectly elastically. Before the collision, block A has a speed of $4\ \text{m s}^{-1}$ and block B has a speed of $6\ \text{m s}^{-1}$. After the collision, block B moves back along its original path with a speed of $2\ \text{m s}^{-1}$.

Calculate the speed of block A after the collision.

speed = ______ $\text{m s}^{-1}$

State the principle of conservation of momentum.

By considering the components of the momenta along the line AB, calculate $\theta$.

$\theta$ = ______ $^\circ$

By calculation of kinetic energies, state and explain whether the collision of the balls is inelastic or perfectly elastic.