The topic of momentum, impulse, and change is fundamental to understanding mechanics in physics. This post will guide you through worksheet solutions, providing insights into how these concepts interact and apply to real-world problems.
Understanding Momentum
Momentum, often denoted by the symbol p, is a vector quantity defined as the product of an object’s mass (m) and its velocity (v). Here’s how it’s mathematically expressed:
[ p = m \cdot v ]
- Momentum depends on both the mass and the velocity of the object.
- The direction of momentum corresponds to the direction of velocity.
Principles of Momentum Conservation
Momentum conservation occurs in an isolated system, where no external forces act upon the system. In such a system, the total momentum before an interaction equals the total momentum after the interaction, even if there are internal forces:
[ p{initial} = p{final} ]
This principle can be observed in various scenarios:
- Collisions, where objects transfer momentum.
- Explosions, where objects initially at rest move apart with momentum.
- Recoiling, like a gun after firing.
Impulse
Impulse (J) is the change in momentum of an object. It can be calculated using the following formula:
[ J = \Delta p = F \cdot \Delta t ]
- J is impulse in Newton-seconds (Ns).
- F represents the force applied.
- Δt is the time interval over which the force is applied.
Relationship Between Impulse and Momentum Change
The impulse-momentum theorem states that the impulse experienced by an object is equal to the change in its momentum. This theorem is particularly useful for analyzing:
- The effect of short duration, high force impacts like car crashes.
- Longer duration, smaller force applications, such as catching a ball with a baseball glove.
Worksheet Solutions
Problem 1: Car Collision
A 1500 kg car traveling at 20 m/s collides with a stationary 2500 kg car. If the collision is elastic, calculate the velocities of each car after the collision.
⚠️ Note: In an elastic collision, kinetic energy is conserved in addition to momentum.
| Before Collision | Momentum | After Collision | Momentum |
|---|---|---|---|
| Car 1: 1500 kg at 20 m/s | 30,000 kg⋅m/s | Car 1: v1 | 1500v1 kg⋅m/s |
| Car 2: 2500 kg at 0 m/s | 0 kg⋅m/s | Car 2: v2 | 2500v2 kg⋅m/s |
| Total Momentum | 30,000 kg⋅m/s | Total Momentum | 30,000 kg⋅m/s |
Using conservation of momentum and kinetic energy, we derive the following equations:
[ 1500v_1 + 2500v_2 = 30,000 ]
[ 0.5 \times 1500 \times 20^2 = 0.5 \times 1500 \times v_1^2 + 0.5 \times 2500 \times v_2^2 ]
Solving these equations simultaneously:
- Car 1’s velocity after collision: 6.67 m/s
- Car 2’s velocity after collision: 13.33 m/s
Problem 2: Impulse to Stop a Sliding Box
A box of mass 10 kg is sliding at 5 m/s when a worker applies a force to stop it. If the worker applies the force for 2 seconds, calculate the impulse and the force required.
Impulse:
[ J = m \cdot \Delta v = 10 \times (-5) = -50 \text{ kg⋅m/s} ]
Since impulse equals the force times time:
[ F \cdot \Delta t = J ] [ F = \frac{-50}{2} = -25 \text{ N} ]
The negative sign indicates the direction opposite to motion.
FAQ Section
What is the difference between momentum and kinetic energy?
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Momentum is a vector quantity depending on mass and velocity, while kinetic energy is a scalar quantity, only considering the square of velocity and mass.
Why do we use impulse in physics?
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Impulse is used to analyze the effect of forces over time, especially in short, intense interactions where average force over time is significant.
Can momentum be conserved in all types of collisions?
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In an isolated system where no external forces act, momentum is always conserved, whether the collision is elastic or inelastic.
