Momentum – problems and solutions

1. 0.5-kg ball moves at 2 m/s. Then the ball is hit with force F opposite to the ball direction, so the ball speed is changed to 6 m/s. The ball in contact with a hitter for 0.01 second, what is the change in momentum of the ball.

__Known :__

Mass of ball (m) = 0.5 kg

Initial velocity (v_{o}) = 2 m/s

Final velocity (v_{t}) = -6 m/s

Time interval (t) = 0.01 second

__Wanted :__ The change in momentum

__Solution :__

∆p = m v_{t} – m v_{o} = m (v_{t }– v_{o})

∆p = (0.5 kg)(- 6 m/s – 2 m/s)

∆p = (0.5 kg)(-8)

∆p = 4 kg m/s

2. A 100-gram object moves at 5 m/s. Force F works for 0.2 seconds to stop the object. Determine magnitude of force F.

__Known :__

Mass of object (m) = 100 gram = 100/1000 = 0.1 kg

Initial velocity (v_{o}) = 5 m/s

Final velocity (v_{t}) = 0

Time interval (Δt) = 0.2 seconds

__Wanted :__ Magnitude of force F

__Solution :__

I = ΔP

F (Δt) = m (v_{t} – v_{o})

F (0.2) = 0.1 (0 – 5)

F (0.2) = 0.1 (– 5)

F (0.2) = -0,5

F = -0,5 / 0.2

F = -2.5 N

Magnitude of force F = 2.5 Newton. Minus sign indicates the direction of force opposite with the direction of the object.

3. A 100-gram ball free fall from the height of 20 cm without initial velocity and then the ball hits the floor. After collision, the ball is reflected upward (acceleration due to gravity is 10 ms^{-2}). What is the change in momentum of ball.

__Known :__

Mass of ball (m) = 100 gram = 0.1 kg

Height (h) = 20 cm = 0.2 meters

Acceleration due to gravity (g) = 10 m/s^{2}

Velocity of ball after hits the floor (v_{t}) = 1 m/s

__Wanted :__ The change in momentum

__Solution :__

__Velocity of ball before collision ____(v___{o}__)__

Calculate velocity of ball before collision using equation of free fall motion. __Known :__ height of ball (h) = 0.2 meters, acceleration due to gravity (g) = 10 m/s^{2}. __Wanted :__ velocity of ball when hits the floor.

v^{2} = 2 g h

v^{2} = 2 (10)(0.2) = 4

v = √4 = -2 m/s

Minus sign indicates that the direction of ball before collision is opposite with the direction of ball after collision.

__The change in momentum of ball (Δp) __

Δp = m v_{t }– m v_{o} = m (v_{t }– v_{o})

Δp = (0.1)(1 – (-2)) = (0.1)(1 + 2) = (0.1)(3) = 0.3 Newton second

4. An object initially at rest, explosion into 2 parts with a ratio of 3 : 2. The larger part of the mass is thrown at a speed of 20 m/s. What is the velocity of the smaller part.

__Known :__

Mass of object 1 before explosion = m

Velocity of object 1 before explosion = 0 (object at rest)

Mass of the larger part after explosion (m_{1}) = 3m

Mass of the smaller part after explosion (m_{2}) = 2m

Velocity of the larger part after explosion (v_{1}‘) = 20 m/s

__Wanted :__ Velocity of the smaller part after explosion (v_{2}‘)

__Solution :__

The equation of the law of the conservation of momentum :

m_{1 }v_{1 }= m_{1 }v_{1}‘ + m_{2 }v_{2}‘

(m)(0) = (3m)(20) + (2m) v_{2}‘

0 = 60m + (2m) v_{2}‘

60m = -2m v_{2}‘

60 = -2 v_{2}‘

v_{2}‘ = -60/2

v_{2}‘ = -30 m/s

Minus sign indicates that the direction of the smaller part is opposite with the direction of the larger part.

**What is momentum?****Answer:**Momentum is a vector quantity that represents the product of an object’s mass and its velocity. It indicates both the direction and magnitude of motion and is often symbolized by $p$, defined as $p=mv$.

**How is momentum different from velocity?****Answer:**While both momentum and velocity are vector quantities related to motion, velocity solely describes the speed and direction of an object’s motion, whereas momentum considers both the object’s mass and its velocity. Momentum provides a measure of how difficult it is to stop a moving object.

**What is the principle of conservation of momentum?****Answer:**The principle of conservation of momentum states that in an isolated system, the total momentum before an event (like a collision) must be equal to the total momentum after the event, provided no external forces act on the system.

**How does impulse relate to momentum?****Answer:**Impulse is the change in momentum of an object when a force is applied over a specific time interval. It’s given by the product of the force and the time over which it acts, and it equals the change in momentum: $Impulse=F×Δt=Δp$.

**Why do airbags in cars help in preventing injuries during a collision?****Answer:**Airbags increase the time over which the force acts on an occupant during a collision. This reduces the average force experienced by the occupant. By the impulse-momentum relationship, spreading out the change in momentum over a longer time reduces the force, helping to prevent injuries.

**What is the difference between elastic and inelastic collisions in terms of momentum and kinetic energy?****Answer:**In both elastic and inelastic collisions, momentum is conserved. However, in elastic collisions, total kinetic energy is also conserved, while in inelastic collisions, it is not.

**How is momentum conserved in the motion of recoil of a gun when it’s fired?****Answer:**When a gun is fired, the bullet moves forward with a certain momentum. To conserve momentum in the isolated system (assuming external forces like air resistance are negligible), the gun must move backward with an equal and opposite momentum. This backward motion is known as recoil.

**Why do objects of different masses fall at the same rate in a vacuum but have different momenta?****Answer:**In a vacuum, the acceleration due to gravity is the same for all objects regardless of their mass. So, different masses fall at the same rate. However, momentum depends on both velocity and mass. Since momentum is $p=mv$, two objects with different masses but the same velocity will have different momenta.

**How does Newton’s third law relate to conservation of momentum?****Answer:**Newton’s third law states that for every action, there’s an equal and opposite reaction. When two objects interact, they exert equal and opposite forces on each other for the same time, meaning they experience equal and opposite changes in momentum. This results in the total momentum before the interaction being equal to the total momentum after, thus conserving momentum.

**Why is it easier to stop a moving bicycle than a moving car, even if they’re traveling at the same speed?**

**Answer:**While the bicycle and car might have the same velocity, their momenta are different because of their different masses. Momentum depends on both velocity and mass. A car, being much more massive than a bicycle, has a greater momentum at the same speed. Thus, a greater impulse (or change in momentum) is required to stop the car compared to the bicycle.