Linear Momentum and Impulse: Principles, Applications, and Worked Examples

Linear Momentum

Definition and Properties

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses due to its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.

• Linear momentum (p) is defined for an object moving in one dimension as the product of its mass (m) and velocity (v).

• The formula for linear momentum is: $p = mv$

• Units: kilogram meter per second (kg m s-1).

• Momentum is conserved in a system unless acted upon by an external resultant force.

Where:

• p: momentum (kg m s-1)

• m: mass (kg)

• v: velocity (m s-1)

Direction of Momentum

Momentum is a vector quantity, so it is characterized by both magnitude and direction.

• The direction of motion is usually assigned as positive; the opposite direction is negative.

• Example: If a ball of mass 60 g travels at 2 m s-1, its momentum is $p = 0.12$ kg m s-1.

• If the ball rebounds in the opposite direction at the same speed, its momentum is $-0.12$ kg m s-1.

Conservation of Linear Momentum

Principle and Application

The principle of conservation of linear momentum states that the total linear momentum of a closed system remains constant if no external resultant force acts on the system.

• Conservation Law: $\text{Total momentum before} = \text{Total momentum after}$

• Opposing vectors can cancel each other out, resulting in a net momentum of zero.

• Momentum is always conserved in collisions and explosions (provided no external force acts).

Worked Example: Two Objects Colliding

• Initial velocity of A: $u_A$

• Initial velocity of B: $u_B$

• Final velocity of A: $v_A$

• Final velocity of B: $v_B$

Conservation equation: $m_A u_A + m_B u_B = m_A v_A + m_B v_B$

Worked Example: Tennis Ball and Brick

• Tennis ball: $m = 0.06$ kg, $v = 75$ m s-1

• Brick: $m = 3$ kg, $v = 1.5$ m s-1

• Momentum for both: $p = mv = 4.5$ kg m s-1

• Although both have the same momentum, the brick is much heavier and travels slower than the ball.

Worked Example: Trolley Collision

• Trolley A: $m = 80$ kg, $v = 3$ m s-1

• Trolley B: $m = 160$ kg, $v = 0$ m s-1

• After collision (stick together): $p_{\text{before}} = 80 \times 3 = 240$ kg m s-1 $p_{\text{after}} = (80 + 160) v$

• Solving: $240 = 240 v \Rightarrow v = 1$ m s-1

Worked Example: Car and Van Collision

• Car: $m = 990$ kg, $v = 2$ m s-1

• Van: $m = 4200$ kg, $v = 0$ m s-1

• After collision, both move together: $p_{\text{before}} = 990 \times 2 = 1980$ kg m s-1 $p_{\text{after}} = (990 + 4200) v$

• Solving: $1980 = 5190 v \Rightarrow v = 0.38$ m s-1

Examiner Tips and Tricks

• Draw diagrams to keep track of masses, velocities, and directions.

• Label all values clearly to avoid mistakes in conservation calculations.

Impulse and Change in Momentum

Definition and Formula

Impulse is the product of the force applied to an object and the time interval over which it acts. It is equal to the change in momentum of the object.

• Impulse formula: $J = F \Delta t$

• Units: newton seconds (N s)

• Impulse is a vector quantity and always acts in the direction of the resultant force.

• A small force acting over a long time has the same effect as a large force acting over a short time.

Impulse-Momentum Theorem

• Newton's second law in terms of momentum: $F = \frac{\Delta p}{\Delta t}$

• Change in momentum: $\Delta p = p_f - p_i$

• If mass is constant, $F = ma$

Impulse Examples

• Rain and hail on an umbrella: Hailstones have greater mass and bounce back, resulting in a greater change in momentum and impulse than raindrops.

• Cricket fielding: Moving hands backward while catching a ball increases the time over which the impulse acts, reducing the force and risk of injury.

Worked Example: Cricket Ball and Racket

• Initial velocity: $u = 30$ m s-1

• Final velocity: $v = -20$ m s-1

• Mass: $m = 0.058$ kg

• Impulse: $J = m(v - u) = 0.058 \times (-20 - 30) = -2.9$ N s

• Direction: Negative sign indicates impulse is to the left.

Examiner Tips and Tricks

• Always check the direction of velocity and impulse; a change in direction must be reflected in the sign.

• Units: $1$N s$= 1$ kg m s-1

Summary Table: Key Equations and Units

Quantity	Symbol	Formula	Units
Linear Momentum	p	$p = mv$	kg m s-1
Impulse	J	$J = F \Delta t$	N s (kg m s-1)
Change in Momentum	$\Delta p$	$\Delta p = p_f - p_i$	kg m s-1
Force (from momentum)	F	$F = \frac{\Delta p}{\Delta t}$	N (kg m s-2)

Additional info:

• All examples and applications are relevant to introductory college-level physics, focusing on mechanics and Newtonian motion.

• Worked examples illustrate the use of conservation of momentum and impulse in real-world scenarios such as collisions and sports.