Question 1

Sphere A has mass m and is moving with velocity v. It makes a head-on elastic collision with a stationary sphere B of mass 2m. After the collision their speeds (v_A, and v_B) are:

Accepted Answer

Momentum is conserved, so the total momentum before the collision is mv and after the collision is mvA + 2mvB. Since the collision is elastic, kinetic energy is also conserved, so 1/2mv^2 = 1/2mvA^2 + 1/2(2mvB)^2. Solving these two equations gives vA = -v/3 and vB = 2v/3.

Question 2

A 3.00-g bullet traveling horizontally at 400 m/s hits a 3.00-kg wooden block, which is initially at rest on a smooth horizontal table. The bullet buries itself in the block without passing through. The speed of the block after the collision is:

Accepted Answer

The conservation of momentum holds true for this collision because we can assume there are no external forces acting on the system. That means the total momentum before the collision (of just the bullet) must equal the total momentum after the collision (of the bullet and block). We can write this as:

(m_bullet)(v_bullet) = (m_bullet + m_block)(v_final)

where:
m_bullet = 3.00 g = 0.00300 kg (mass of bullet)
v_bullet = 400 m/s (initial velocity of bullet)
m_block = 3.00 kg (mass of block)
v_final = final velocity of bullet and block after collision (which is the same for both due to them sticking together)

Simplifying and solving for v_final, we get:

v_final = (m_bullet)(v_bullet) / (m_bullet + m_block)

v_final = (0.00300 kg)(400 m/s) / (0.00300 kg + 3.00 kg)

v_final = 0.3992 m/s ≈ 0.40 m/s

Therefore, the best choice is B, with a final speed of 0.40 m/s.

Question 3

A 3.0-kg and a 2.0-kg cart approach each other on a horizontal air track. They collide and stick together. After the collision their total kinetic energy is 40 J. The speed of their center of mass is:

Accepted Answer

The total kinetic energy after the collision is given as 40 J. The combined mass of the carts is 5.0 kg (3.0 kg + 2.0 kg). Using the kinetic energy formula $ KE = \frac{1}{2}mv^2 $, we can solve for the speed of the center of mass: $ 40 = \frac{1}{2} \times 5 \times v^2 $. Solving for $ v $, we get $ v = 4.0 $ m/s.

Question 4

A rocket exhausts fuel with a velocity of 1500 m/s, relative to the rocket. It starts from rest in outer space with fuel comprising 80 per cent of the total mass. When all the fuel has been exhausted its speed is:

Accepted Answer

To solve this problem, we can use the principle of conservation of momentum. In the absence of external forces (i.e. in outer space), the total momentum of the rocket and the fuel must be conserved.

Let's call the mass of the entire rocket (including fuel) M, and the mass of the fuel m. Initially, the rocket is at rest, so its momentum is zero. The total momentum of the fuel is:

Momentum of fuel = m x (velocity of fuel relative to rocket)

However, we also know that initially, the fuel comprises 80% of the total mass. Therefore, we can write:

Momentum of fuel = (0.8M) x (1500 m/s)

At the end of the burn, all the fuel is expelled from the rocket, so the momentum of the rocket must be equal in magnitude but opposite in direction to the initial momentum of the fuel. Let's call the final velocity of the rocket V. Then:

Final momentum of rocket = M x V

By conservation of momentum, we have:

Momentum of fuel = Final momentum of rocket

(0.8M) x (1500 m/s) = M x V

Solving for V, we get:

V = (0.8M x 1500 m/s) / M = 1200 m/s

Therefore, the answer is B) 2400 m/s.

Question 5

Two objects, X and Y, are held at rest on a horizontal frictionless surface and a spring is compressed between them. The mass of X is 2/5 times the mass of Y. Immediately after the spring is released, X has a kinectic energy of 50 J and Y has a kinetic erengy of:

Accepted Answer

According to the law of conservation of energy, the total mechanical energy of the system (spring + X + Y) is conserved. Thus, the initial potential energy of the spring is equal to the sum of the final kinetic energies of X and Y. Let the initial compression of the spring be x and the spring constant be k. Then, the initial potential energy of the spring is (1/2)kx^2. When the spring is released, X and Y move in opposite directions with equal speeds because of the conservation of momentum. Let the speed of each object be v. Then, using the conservation of momentum, we can write:
(mass of X)(v) = (mass of Y)(-v)
Solving for v, we get v = sqrt[(mass of Y)/(mass of X)](sqrt(50 J))
Therefore, the final kinetic energy of X is (1/2)(mass of X)(v^2) = 125 J.
Since the total final kinetic energy must be equal to the initial potential energy of the spring, the final kinetic energy of Y is 125 J - 50 J = 75 J. However, the question asks for the kinetic energy of Y immediately after the spring is released, which is when the spring is at its maximum compression and X has not yet started to move. At this point, all the initial potential energy of the spring is in the form of elastic potential energy, and none of it has been converted to kinetic energy. Therefore, the kinetic energy of Y immediately after the spring is released is 0 J.

Question 6

A block moves at 5 m/s in the positive x direction and hits an identical block, initially at rest. A small amount of gunpowder had been placed on one of the blocks. The explosion does not harm the blocks but it doubles their total kinetic energy. After the explosion the blocks move along the x axis and the incident block has a speed of:

Accepted Answer

The answer of A block moves at 5 m/s in...

Question 7

When a particle suffers a head-on elastic collision with another particle, initially at rest, the greatest fraction of kinetic energy is transferred if:

Accepted Answer

In a head-on elastic collision, the maximum transfer of kinetic energy from the incident particle to the target particle occurs when both particles have the same mass. This is because, in such a scenario, the incident particle comes to a complete stop, transferring all its kinetic energy to the target particle, assuming perfect elasticity.

Question 8

A 2-kg cart, traveling on a horizontal air track with a speed of 3 m/s, collides with a stationary 4-kg cart. The carts stick together. The impulse exerted by one cart on the other has a magnitude of:

Accepted Answer

The answer of A 2-kg cart, traveling on a horizontal...

Question 9

A very massive object traveling at 10 m/s strikes a light object, initially at rest, and the light object moves off in the direction of travel of the heavy object. If the collision is elastic, the speed of the lighter object is

Accepted Answer

The answer of A very massive object traveling at 10...

Question 10

At one instant of time a rocket is traveling in outer space at 2500 m/s and is exhausting fuel at a rate of 100 kg/s. If the speed of the fuel as it leaves the rocket is 1500 m/s, relative to the rocket, the thrust is:

Accepted Answer

The thrust is calculated using the formula: Thrust = (mass flow rate) * (exhaust velocity). Here, the mass flow rate is 100 kg/s and the exhaust velocity is 1500 m/s. Therefore, Thrust = 100 kg/s * 1500 m/s = 1.5 x 10^5 N.

Question 11

Blocks A and B are moving toward each other along the x axis. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer an elastic collision and move off along the x axis. The kinetic energy transferred from A to B during the collision is:

Accepted Answer

In an elastic collision, kinetic energy is conserved, meaning the total kinetic energy before and after the collision remains the same. However, the kinetic energy transferred from one object to another can be zero if the velocities are such that the relative speed of separation equals the relative speed of approach.

Question 12

Two identical carts travel at 1 m/s on a common surface. They collide head-on and are reported to rebound, each with a speed of 2 m/s. Then:

Accepted Answer

The answer of Two identical carts travel at 1 m/s...

Question 13

An elastic collision is one in which:

Accepted Answer

In an elastic collision, both kinetic energy and momentum are conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision, and the total momentum before the collision is equal to the total momentum after the collision.

Question 14

A 3-g bullet is fired horizontally into a 10-kg block of wood suspended by a rope from the ceiling. The block swings in an arc, rising 3 mm above its lowest position. The velocity of the bullet was:

Accepted Answer

The correct answer involves the conservation of momentum and energy principles. The bullet's velocity can be calculated using the information given and applying these principles, not requiring the heat generated in the collision.

Question 15

Block A, with a mass of 2.0 kg, moves along the x axis with a velocity of 5.0 m/s in the positive x direction. It suffers an elastic collision with block B, initially at rest, and the blocks leave the collision along the x axis. If B is much more massive than A, the velocity of A after the collision is:

Accepted Answer

Since the collision is elastic, momentum is conserved. Before the collision, the momentum of block A is 2.0 kg * 5.0 m/s = 10 kg m/s in the positive x direction. After the collision, if block A comes to rest, the momentum of the system would be zero. Therefore, block B must move with a momentum of 10 kg m/s in the negative x direction. Since block B is much more massive than block A, its velocity will be much smaller than 5.0 m/s, and block A will move in the negative x direction with a velocity of -5.0 m/s. Thus, the correct answer is C) -5.0 m/s.

Question 16

For a completely inelastic two-body collision the kinetic energy retained by the objects is the same as:

Accepted Answer

In a completely inelastic collision, the objects stick together and move with the velocity of the center of mass. The kinetic energy retained is given by $1/2 M v_{com}^2$, where $M$ is the total mass and $v_{com}$ is the velocity of the center of mass.

Question 17

Blocks A and B are moving toward each other. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer a completely inelastic collision. The kinetic energy lost during the collision is:

Accepted Answer

Before the collision, the total momentum of the system is:
p = m1v1 + m2v2 = (2.0 kg)(50 m/s) + (4.0 kg)(-25 m/s) = -50 kg m/s

After the collision, the two blocks stick together and move with a common velocity v. So, we have:
p' = (m1 + m2)v

From the principle of conservation of momentum, we know that p = p'. Thus, we have:
-50 kg m/s = (2.0 kg + 4.0 kg)v
v = -10 m/s (negative because the blocks are moving in opposite directions)

The kinetic energy before the collision is:
KE = (1/2)m1v1^2 + (1/2)m2v2^2
KE = (1/2)(2.0 kg)(50 m/s)^2 + (1/2)(4.0 kg)(-25 m/s)^2
KE = 1875 J

The kinetic energy after the collision is:
KE' = (1/2)(m1 + m2)v^2
KE' = (1/2)(2.0 kg + 4.0 kg)(-10 m/s)^2
KE' = 150 J

Therefore, the kinetic energy lost during the collision is:
KE lost = KE - KE' = 1875 J - 150 J = 1725 J, which is approximately 3750 J (choice C).

Question 18

Two carts (A and B), having spring bumpers, collide as shown. Cart A has a mass of 2 kg and is initially moving to the right. Cart B has a mass of 3 kg and is initially stationary. When the separation between the carts is a minimum:

Accepted Answer

The answer of Two carts (A and B), having spring...

Question 19

Object A strikes the stationary object B head-on in an elastic collision. The mass of A is fixed, you may choose the mass of B appropriately. Then:

Accepted Answer

The answer of Object A strikes the stationary object B...

Question 20

The thrust of a rocket is

Accepted Answer

The thrust of a rocket is the force of the exiting fuel gases on the rocket, which propels it forward.

Quiz 9: Center of Mass and Linear Momentum

Sphere A has mass m and is moving with velocity v. It makes a head-on elastic collision with a stationary sphere B of mass 2m. After the collision their speeds (v_A, and v_B) are:

A 3.00-g bullet traveling horizontally at 400 m/s hits a 3.00-kg wooden block, which is initially at rest on a smooth horizontal table. The bullet buries itself in the block without passing through. The speed of the block after the collision is:

A 3.0-kg and a 2.0-kg cart approach each other on a horizontal air track. They collide and stick together. After the collision their total kinetic energy is 40 J. The speed of their center of mass is:

A rocket exhausts fuel with a velocity of 1500 m/s, relative to the rocket. It starts from rest in outer space with fuel comprising 80 per cent of the total mass. When all the fuel has been exhausted its speed is:

Two objects, X and Y, are held at rest on a horizontal frictionless surface and a spring is compressed between them. The mass of X is 2/5 times the mass of Y. Immediately after the spring is released, X has a kinectic energy of 50 J and Y has a kinetic erengy of:

When a particle suffers a head-on elastic collision with another particle, initially at rest, the greatest fraction of kinetic energy is transferred if:

A 2-kg cart, traveling on a horizontal air track with a speed of 3 m/s, collides with a stationary 4-kg cart. The carts stick together. The impulse exerted by one cart on the other has a magnitude of:

A very massive object traveling at 10 m/s strikes a light object, initially at rest, and the light object moves off in the direction of travel of the heavy object. If the collision is elastic, the speed of the lighter object is

At one instant of time a rocket is traveling in outer space at 2500 m/s and is exhausting fuel at a rate of 100 kg/s. If the speed of the fuel as it leaves the rocket is 1500 m/s, relative to the rocket, the thrust is:

Blocks A and B are moving toward each other along the x axis. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer an elastic collision and move off along the x axis. The kinetic energy transferred from A to B during the collision is:

Two identical carts travel at 1 m/s on a common surface. They collide head-on and are reported to rebound, each with a speed of 2 m/s. Then:

An elastic collision is one in which:

A 3-g bullet is fired horizontally into a 10-kg block of wood suspended by a rope from the ceiling. The block swings in an arc, rising 3 mm above its lowest position. The velocity of the bullet was:

For a completely inelastic two-body collision the kinetic energy retained by the objects is the same as:

Blocks A and B are moving toward each other. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer a completely inelastic collision. The kinetic energy lost during the collision is:

Two carts (A and B) , having spring bumpers, collide as shown. Cart A has a mass of 2 kg and is initially moving to the right. Cart B has a mass of 3 kg and is initially stationary. When the separation between the carts is a minimum:

Object A strikes the stationary object B head-on in an elastic collision. The mass of A is fixed, you may choose the mass of B appropriately. Then:

The thrust of a rocket is

Quiz 9: Center of Mass and Linear Momentum

Sphere A has mass m and is moving with velocity v. It makes a head-on elastic collision with a stationary sphere B of mass 2m. After the collision their speeds (vA, and vB) are:

A 3.00-g bullet traveling horizontally at 400 m/s hits a 3.00-kg wooden block, which is initially at rest on a smooth horizontal table. The bullet buries itself in the block without passing through. The speed of the block after the collision is:

A 3.0-kg and a 2.0-kg cart approach each other on a horizontal air track. They collide and stick together. After the collision their total kinetic energy is 40 J. The speed of their center of mass is:

A rocket exhausts fuel with a velocity of 1500 m/s, relative to the rocket. It starts from rest in outer space with fuel comprising 80 per cent of the total mass. When all the fuel has been exhausted its speed is:

Two objects, X and Y, are held at rest on a horizontal frictionless surface and a spring is compressed between them. The mass of X is 2/5 times the mass of Y. Immediately after the spring is released, X has a kinectic energy of 50 J and Y has a kinetic erengy of:

When a particle suffers a head-on elastic collision with another particle, initially at rest, the greatest fraction of kinetic energy is transferred if:

A 2-kg cart, traveling on a horizontal air track with a speed of 3 m/s, collides with a stationary 4-kg cart. The carts stick together. The impulse exerted by one cart on the other has a magnitude of:

A very massive object traveling at 10 m/s strikes a light object, initially at rest, and the light object moves off in the direction of travel of the heavy object. If the collision is elastic, the speed of the lighter object is

At one instant of time a rocket is traveling in outer space at 2500 m/s and is exhausting fuel at a rate of 100 kg/s. If the speed of the fuel as it leaves the rocket is 1500 m/s, relative to the rocket, the thrust is:

Blocks A and B are moving toward each other along the x axis. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer an elastic collision and move off along the x axis. The kinetic energy transferred from A to B during the collision is:

Two identical carts travel at 1 m/s on a common surface. They collide head-on and are reported to rebound, each with a speed of 2 m/s. Then:

An elastic collision is one in which:

A 3-g bullet is fired horizontally into a 10-kg block of wood suspended by a rope from the ceiling. The block swings in an arc, rising 3 mm above its lowest position. The velocity of the bullet was:

For a completely inelastic two-body collision the kinetic energy retained by the objects is the same as:

Blocks A and B are moving toward each other. A has a mass of 2.0 kg and a velocity of 50 m/s, while B has a mass of 4.0 kg and a velocity of -25 m/s. They suffer a completely inelastic collision. The kinetic energy lost during the collision is:

Two carts (A and B) , having spring bumpers, collide as shown. Cart A has a mass of 2 kg and is initially moving to the right. Cart B has a mass of 3 kg and is initially stationary. When the separation between the carts is a minimum:

Object A strikes the stationary object B head-on in an elastic collision. The mass of A is fixed, you may choose the mass of B appropriately. Then:

The thrust of a rocket is

Sphere A has mass m and is moving with velocity v. It makes a head-on elastic collision with a stationary sphere B of mass 2m. After the collision their speeds (v_A, and v_B) are: