Evans SPH 4U1

Unit 4: Energy & Momentum

Note 3: Linear Momentum and Impulse

Reference: Chapter 5.1 & 5.2

Linear Momentum & Impulse:

Linear momentum is a physical quantity that represents the product of the mass of an object and its instantaneous velocity (its in the same direction of the velocity vector).

SI unit: kgm/s

Example:

A 2.0 kg mass is travelling, initially at 2.0 m/s [E]. Its final velocity is recorded as 3.0 m/s [W]. Calculate the change in momentum.

Impulse:

Whenever a force acts on an object (usually for a short time interval) we say an impulse acts on the object. Impulse is defined as the prodduct of the applied force and the time interval for which the force acts.  It, too, is a vector quantity. The direction of the impulse vector is the same as the force vector.

SI unit: Ns

Area under an

Example:

A force of 1.0 x 103 N[E] acts on an object for 20ms. calculate the impulse due to this force. (Answer: 20 Ns[E])

In cases where the applied force varies in magnitude the easy solution is to draw, if possible, a force vs. time graph. The area under the force time graph is a measure of the impulse. (note: calculus students could solve using integration between limits).

Derive the Relationship between Impulse and Linear Momentum:

Example:

A 0.25 kg ball accelerates uniformly from a velocity of 2.0 m/s [E] to a velocity of 14 m/s [E] in 4.0 seconds. Calculate the unbalanced force acting on the ball using 2 different methods.

Note: Newton's 2nd Law, F = ma, doesn't hold for small particles near the speed of light but the more general form does.

The Principle of Conservation of Linear Momentum:

This principle states that in all isolated systems (no net external force), the total linear momentum stays unchanged. In other words, if you were to measure the total linear momentum BEFORE an event in an isolated system, it should be exactly equal to the total linear momentum AFTER the event.

Example 1:

A stationary firecracker of mass 2.0 kg spontaneously splits into 2 segments. Fragment x flies off to the east at 3.0 m/s and has a mass of 0.5 kg. Calculate the velocity with which the other fragment is propelled.  Assume that no external forces act on the body.

Example 2:

A 100 g glider on an air track is travelling at 1.0 m/s towards a stationary glider of mass 300g. On impact, the gliders couple to each other. Calculate the final velocity of the glider. Answer: 0.25 m/s [forward]

Example 3:

A stationary mass of 21kg splits spontaneously into 3 equally massive fragments. Fragment A is observed to fly off horizontally at 3.0 m/s[E]. Fragment B takes off again horizontally, at 4.0 m/s[S]. Calculate the momentum and velocity of fragment C.

Answer: momentum is 35 kgm/s[N37W], velocity 5.0m/s[n37w]

Example 4:

Two students of mass 60kg each standing on a cart with a mass of 40 kg jump off the cart with a velocity of 6m/s relative to the cart. Assuming no friction calculate the velocity of the cart after both students have jumped off the cart. Answer: 4.5 m/s

Example 5:

A stationary mass of 21kg splits spontaneously into 3 equally massive fragments. Fragment A is observed to fly off horizontally at 3.0 m/s[E]. Fragment B takes off again horizontally, at 4.0 m/s[S]. Calculate the momentum and velocity of fragment C.

Answer: momentum is 35 kgm/s[N37W], velocity 5.0m/s[n37w]

Page 237 #7, 8

Page 245 #5, 6, 7

Review Questions for Chapter 4 Page 227 # 11, 12, 13, 16, 17, 22, 23