All questions of Linear Momentum and Impulse for EmSAT Achieve Exam

Understanding Kinetic Energy and Momentum
Kinetic energy (KE) and momentum (p) are two fundamental concepts in mechanics. Their relationships can help us understand the impact of changes in one on the other.
Kinetic Energy Formula
- Kinetic energy is given by the formula: KE = (1/2)mv^2, where m is mass and v is velocity.
- If the kinetic energy increases by 300%, this means the new kinetic energy is 4 times the original (original + 3 times original).
Momentum Formula
- Momentum is given by the formula: p = mv.
- Momentum depends directly on mass and velocity.
Analyzing the Increase in Kinetic Energy
- Let the initial kinetic energy be KE1 = (1/2)mv1^2.
- The new kinetic energy after a 300% increase: KE2 = 4 * KE1 = 4 * (1/2)mv1^2 = 2mv1^2.
Relating Kinetic Energy to Momentum
- To find the new momentum, we first need to express the new velocity (v2).
- From KE2 = (1/2)mv2^2, we can deduce that: 2mv1^2 = (1/2)mv2^2, leading to v2^2 = 4v1^2, hence v2 = 2v1.
Calculating the Increase in Momentum
- Initial momentum: p1 = mv1.
- New momentum: p2 = mv2 = m(2v1) = 2mv1 = 2p1.
- The increase in momentum: Δp = p2 - p1 = 2p1 - p1 = p1.
Percentage Increase in Momentum
- The percentage increase in momentum is given by: (Δp / p1) * 100 = (p1 / p1) * 100 = 100%.
Thus, the corresponding increase in momentum is 100%, which is option B.

Explanation:

Given Data:
- Mass of each sphere, m = 10 kg
- Initial velocities, u1 = 15 m/s and u2 = 5 m/s

Loss in Kinetic Energy:
- The initial total kinetic energy of the system is given by:
KE_initial = (1/2) * m1 * u1^2 + (1/2) * m2 * u2^2
= (1/2) * 10 * 15^2 + (1/2) * 10 * 5^2
= 1125 + 125
= 1250 J
- After the collision, the final total kinetic energy of the system is given by:
KE_final = (1/2) * (m1 + m2) * V^2
where V is the common final velocity of both spheres.
- Since the collision is inelastic, kinetic energy is not conserved. The loss in kinetic energy is given by:
Loss in KE = KE_initial - KE_final
- From the law of conservation of momentum:
m1 * u1 + m2 * u2 = (m1 + m2) * V
10 * 15 + 10 * 5 = 20 * V
150 + 50 = 20V
V = 10 m/s
- Substituting the value of V in the equation for KE_final:
KE_final = (1/2) * 20 * 10^2
= 1000 J
- Therefore, the loss in kinetic energy is:
Loss in KE = 1250 - 1000
= 250 J
Therefore, the correct answer is option C, 250 Nm.

Impulse during Contact:
In order to calculate the impulse during the contact between the ball and the floor, we can use the principle of conservation of momentum. The impulse experienced by an object is equal to the change in momentum it undergoes during a collision.

Initial Momentum:
The initial momentum of the ball can be calculated using the formula:
\[p_{initial} = m \times v_{initial}\]
where
m = mass of the ball (1.0 kg)
v_{initial} = initial speed of the ball (25 m/s)

Final Momentum:
The final momentum of the ball can be calculated using the formula:
\[p_{final} = m \times v_{final}\]
where
m = mass of the ball (1.0 kg)
v_{final} = final speed of the ball (10 m/s)

Impulse:
The impulse experienced by the ball during contact is given by the formula:
\[Impulse = p_{final} - p_{initial}\]
Substitute the values into the formula to find the impulse:
\[Impulse = (1.0 \times 10) - (1.0 \times 25) = 10 - 25 = -15\,kg\,m/s\]
The negative sign indicates that the direction of the impulse is opposite to the initial direction of motion. Therefore, the magnitude of the impulse is 15 N-s.
Therefore, the correct answer is option C which is 35 N-s.

Conservation of Momentum in Elastic and Inelastic Collisions
In both elastic and inelastic collisions, momentum is conserved. This means that the total momentum of the system before the collision is equal to the total momentum of the system after the collision.

Elastic Collision
- In an elastic collision, kinetic energy is also conserved along with momentum.
- The objects bounce off each other without any loss of kinetic energy.
- The total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Inelastic Collision
- In an inelastic collision, kinetic energy is not conserved.
- Some of the kinetic energy is transformed into other forms of energy like sound, heat, or deformation of the objects.
- Despite this loss of kinetic energy, momentum is still conserved in the system.

Explanation
- Momentum is conserved in collisions because there is no external force acting on the system during the collision.
- According to Newton's third law of motion, the forces between the objects in the collision are equal and opposite, resulting in a constant total momentum.
- This conservation of momentum is a fundamental principle in physics and is used to analyze and predict the behavior of objects in collisions.
Therefore, in both elastic and inelastic collisions, momentum is conserved, making it a crucial concept in understanding the dynamics of interactions between objects.