Impulse

**Collision** – is a short duration interaction between two
objects

**Impulsive force** – a large force exerted
during a small interval of time

Graphically the impulsive force may look like the graph below:

The area under the curve (integral in Calculus)
is the **impulse ( J)**.

Example 1

When a male bighorn sheep runs head-first into another male, the rate at which its speed drops to zero is dramatic. A graph is shown. The peak acceleration has a magnitude of and the duration of the collision is 0.27 s. Assume that the sheep's mass is 90.0 kg. What are the magnitudes of the impulse and average force due to the collision?

Example 2

A person
stands under an umbrella during a rain shower. A few minutes later the
rain drops turn to hail–though the number of "drops"
hitting the umbrella per time and the speed remains the same.
Is the force required to hold the umbrella in the hail **(a)** the
same as **(b)** more than, or **(c)** less
than the force required in the rain?

Momentum and the Impulse–Momentum Theorem

Deriving Momentum

Example 3

After winning a prize on a game show, a 72 kg contestant jumps for joy. (a) If the jump results in an upward speed of 2.1 m/s, what is the impulse experienced by the contestant? (b) Before the jump, the floor exerts an upward force of mg on the contestant. What additional average upward force does the floor exert if the contestant pushes down on it for 0.36 sec during the jump?

Example 4

A 0.144 kg baseball is moving toward home plate with a speed of 43.0 m/s when it is bunted (hit softly). The bat exerts an average force of 6500 N on the ball for 1.30 ms. The average force is directed toward the pitcher, which we take to be positive x direction. What is the final speed of the ball?

Example 5

Far in space, where gravity is negligible, a 425 kg rocket traveling at 75 m/s fires its engine. The figure shows the thrust force as a function of time. The mass lost by the rocket during these 30.0 s is negligible.

a. What impulse does the engine impart to the rocket?

b. At what time does the rocket reach its maximum speed? What is the maximum speed?

Conservation of Momentum

**Law of Conservation
of Momentum** – The total
momentum P of an isolated system is a constant. Interactions within the
system do not change the system's total momentum.

Example 6

Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in canoe 1 pushes on canoe 2 with a force of 46 N to separate the canoes. If the mass of canoe 1 and its occupants is 130 kg, and the mass of canoe 2 and its occupants is 250 kg, find the momentum of each canoe after 1.20 s of pushing.

Example 7

A honeybee with a mass of 0.150 g lands on one end of a floating 4.75 g popsicle stick. After sitting at rest for a moment, it runs toward the other end with a velocity relative to the still water. The stick moves in the opposite direction with a speed of 0.120 cm/s. What is the velocity of the bee? (Let the direction of the bee's motion be the positive x direction.)

Inelastic Collisions

**Perfectly inelastic
collisions** – a collision
when the two objects stick together and move with a common final velocity.

Example 8

A 1200 kg car moving at 2.5 m/s is struck in the rear by a 2600 kg truck moving at 6.2 m/s. If the vehicles stick together after the collision, what is their speed immediately after colliding?

Example 9

On a touchdown attempt, a 95.0 kg running back runs toward the end zone at 3.75 m/s. A 111 kg linebacker moving at 4.10 m/s meets the runner in a head-on collision. If the two players stick together, what is their velocity immediately after the collision?

Example 10

A 5o kg archer, standing on frictionless ice, shoots a 100 g arrow at a speed of 100 m/s. What is the recoil speed of the archer?

Example 11

The parking brake on a 2000 kg Cadillac has failed, and it is rolling slowly, at 1 mph, toward a group of small innocent children. As you see the situation, you realize there is just time for you to drive your 1000 kg Volkswagen head-on into the Cadillac and thus to save the children. With what speed should you impact the Cadillac to bring it to a halt?

Example 12

Connor is gliding on his skateboard at 4 m/s. He suddenly jumps backward off the skateboard, kicking the skateboard forward at 8 m/s. How fast is Connor going as his feet hit the ground? Connors mass is 50 kg and the skateboard's mass is 5 kg.

Momentum and Collisions in Two Dimensions

Example 13

A firecracker placed inside a coconut of mass M, initially at rest on a frictionless floor, blows the coconut into three pieces that slide across the floor. See the figure below. Piece C, with mass 0.30M, has final speed 5 m/s.

(a) What is the speed of piece B, with mass 0.20M?

(b) What is the speed of piece A?

Example 14

A car with a mass of 950 kg and a speed of 16 m/s approaches an intersection, as shown. A 1300 kg minivan traveling at 21 m/s is heading for the same intersection. The car and minivan collide and stick together. Find the speed and direction of the wrecked vehicles just after the collision, assuming external forces can be ignored.

Example 15

Two skaters collide and embrace, in a completely inelastic collision. Thus, they stick together after impact, as in the figure, where the origin is placed at the point of collision. Alfred, whose mass is 82 kg, is originally moving east with speed 6.2 km/hr. Barbara, whose mass is 55 kg, is originally moving north with speed 7.8 km/hr. What is the velocity of the couple after they collide?

Angular Momentum

**Law of Conservation
of Angular Momentum** – The angular momentum of a rotating object subject to no net torque is
a constant. The final angular momentum is equal to the initial angular
momentum.

Example 16

For a classroom demonstration, a sits on a piano stool holding a sizable mass in each hand. Initially, the student holds his arms outstretched and spins about the axis of the stool with an angular speed of 3.74 rad/s. The moment of inertia in this case is . While still spinning, the student pulls his arms in to his chest, reducing the moment of inertia to . What is the student's angular speed now?

Example 17

A star of radius rotates with an angular speed . If this star collapses to a radius of 20.0 km, find its final angular speed. (Treat the star as if it were a uniform sphere, and assume that no mass is lost as the star collapses.)