0625 · 1.6Momentum

Momentum, mass and velocity

0:00When you catch a fast-moving ball, letting your hands move backwards makes the catch feel gentler. The ball still comes to rest, but its motion changes over a longer time. Momentum explains why that matters. By the end of this video, you will understand collisions, recoil, and the connection between force and stopping time. This topic is Supplement content for the Extended course. You need three familiar ideas before we begin.

0:23Mass measures how much matter an object contains, velocity describes speed in a stated direction, and resultant force means the overall force. We will consider motion along one straight line, so there are only two possible directions. Imagine two trolleys moving at the same velocity, with one carrying more mass. Bringing the heavier trolley to rest requires a greater change in momentum. Now keep the mass unchanged and increase its velocity.

0:47Again, stopping it requires a greater momentum change. Momentum combines these two features of motion into one quantity. Momentum is defined as mass times velocity. The equation is momentum equals mass multiplied by velocity. Doubling the mass doubles the momentum if velocity stays unchanged. Doubling the velocity also doubles the momentum if mass stays unchanged. The unit of momentum is kilogram metres per second, because we multiply kilograms by metres per second.

1:15Consider a trolley of mass two kilograms moving right at three metres per second. Its momentum equals 2 × 3. That gives six kilogram metres per second to the right. Notice that we include the direction, because momentum is a vector quantity. A vector has both magnitude and direction. For calculations, we choose one direction as positive and keep that choice throughout. If right is positive, a leftward velocity is negative.

1:39The same two-kilogram trolley moving left at three metres per second has momentum negative six kilogram metres per second. The minus sign indicates direction, not negative mass or an impossible amount of motion.

Impulse and change in momentum

1:51Momentum does not tell us the force on an object by itself. An object can have momentum while moving at constant velocity with no resultant force. Force is connected to a change in momentum. This distinction matters because a moving object does not need a continuing resultant force simply to keep moving. A force acting over time can change an object's momentum. Impulse is defined as force times the time for which the force acts.

2:14The equation is impulse equals force multiplied by the time interval. Impulse also equals the change in momentum. These are two descriptions of the same interaction, rather than two unrelated facts. The force tells us how quickly momentum changes, while the time tells us how long that change continues. The same force acting for twice as long produces twice the impulse. For an unchanged mass, change in momentum equals mass multiplied by the change in velocity.

2:39The change in velocity is final velocity minus initial velocity. Always subtract the initial momentum from the final momentum. Suppose a resultant force of six newtons acts rightwards on a trolley for zero point five seconds. Impulse equals 6 × 0.5. The impulse is three newton seconds to the right. Therefore, the trolley's momentum increases by three kilogram metres per second to the right. Newton seconds and kilogram metres per second are equivalent units.

3:06For a constant force, we use that force directly in the impulse equation. If the force varies during the interaction, we use its average value over the stated time. The impulse still equals the total change in momentum. This is useful for collisions, where the contact force usually changes while the objects touch.

Rebound forces and stopping time

3:24Now consider a ball of mass zero point two zero kilograms striking a wall. It approaches rightwards at ten metres per second and rebounds leftwards at five metres per second. Choose right as positive, so its final velocity is negative five metres per second. The contact lasts zero point one zero seconds. We will calculate the impulse and average resultant force. The initial momentum equals 0.20 × 10, giving positive two kilogram metres per second.

3:49The final momentum equals 0.20 × −5, giving negative one kilogram metre per second. The change equals negative one minus positive two. Therefore, the impulse is negative three newton seconds, directed leftwards. This rebound shows why direction cannot be ignored. The ball must first lose its rightward momentum and then gain leftward momentum. Subtracting the speeds alone would miss that reversal. The magnitude of its momentum change is three kilogram metres per second, even though its final speed is smaller than its initial speed.

4:20Resultant force is defined as the change in momentum per unit time. The equation is resultant force equals change in momentum divided by the time interval. For our ball, divide negative three by zero point one zero. The average resultant force is negative thirty newtons. That means a force of thirty newtons to the left. This equation also connects momentum to the familiar relationship between force and acceleration. For constant mass, change in momentum equals mass multiplied by change in velocity.

4:49Dividing by time gives mass multiplied by acceleration. So the momentum equation agrees with resultant force equals mass multiplied by acceleration. Return to catching the ball. If its initial motion is unchanged and it finishes at rest, the momentum change is unchanged. Moving your hands backwards increases the stopping time. The same momentum change divided by a longer time gives a smaller average resultant force. It is the stopping time that changes, not the required impulse.

Conservation of momentum

5:15Next, consider two objects interacting together. We call the objects we are studying a system. During a collision, they exert equal and opposite forces on each other for the same time. Their impulses are therefore equal and opposite. One object's momentum change balances the other object's momentum change. The principle of conservation of momentum states that total momentum remains constant when no resultant external force acts on the system. An external force comes from outside the chosen system.

5:44Forces between the two objects are internal forces. They can change each object's momentum without changing the total momentum of the pair. Here is a collision calculation with opposite directions. A two-kilogram trolley moves right at three metres per second. A one-kilogram trolley moves left at two metres per second. They collide and stick together, with negligible external forces during the collision. We choose right as positive and calculate their shared final velocity.

6:09The first trolley's initial momentum equals 2 × 3, giving positive six kilogram metres per second. The second trolley's initial momentum equals 1 × −2, giving negative two kilogram metres per second. Adding them gives total initial momentum of positive four kilogram metres per second. We add signed momenta, not speeds. Conservation means the total final momentum must also be positive four kilogram metres per second. Because the trolleys stick together, their combined mass is three kilograms.

6:39Their shared velocity equals 4 ÷ 3. This gives approximately one point three three metres per second to the right. The positive sign identifies the final direction. Sticking together is not required for momentum conservation. Objects can separate after a collision, and their final momenta must still add to the same total. Momentum conservation does not mean that each object keeps its own momentum. Nor does it mean that the objects must have equal velocities after every collision.

7:06The same principle applies when objects initially at rest push apart, as in an explosion or spring release. Their total initial momentum is zero. With no resultant external force, their total final momentum must also be zero. Their momenta are equal in magnitude and opposite in direction, but their speeds need not be equal. Suppose a two-kilogram trolley moves right at three metres per second after a spring release. Its momentum is positive six kilogram metres per second.

7:32A three-kilogram trolley must therefore have momentum negative six kilogram metres per second. Dividing negative six by three gives negative two metres per second. The heavier trolley moves left at the lower speed. In examinations, begin calculations by choosing a positive direction and identifying the objects in your system. Convert masses to kilograms and times to seconds before substituting. For momentum conservation, write total momentum before equals total momentum after.

7:59For force or impulse, calculate final momentum minus initial momentum, including direction signs throughout. An exam-style explanation might ask why a longer stopping time reduces the average force on a passenger. A precise answer would state that the passenger has the same change in momentum. Increasing the stopping time reduces the change in momentum per unit time. Therefore, the average resultant force is smaller. Each sentence links a physical change to its consequence.

8:25Let us bring the ideas together. Momentum is mass times velocity, and its direction matters. Impulse is force times the time for which it acts, and equals the change in momentum. Resultant force is change in momentum per unit time. Total momentum is conserved when no resultant external force acts on the system.