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Calculate Speed, Momentum, Acceleration, Work, and Power

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### Calculate Speed, Momentum, Acceleration, Work, and Power

4A calculate speed, momentum, acceleration, work, and power in systems such as in the human body, moving toys and machines;

Units

Change in temperature = ∆T = Final Temperature – Initial Temperature

=Tf - Ti

Change in velocity = ∆V = Vf – Vi

Speed

- The distance traveled per unit of time.
- Common units: m/s and km/hr
- Speed = distance
- Time
- Instantaneous speed is speed at a given instant. Just look at speedometer in a car; that is instantaneous speed.

Graphs of constant speed

Spacing or distance covered between equal time intervals is the same.

0m 1m 2m 3m 4m

| | | | |

0s 1s 2s 3s 4s

Slope = ∆y = speed ∆x

Steeper slope = faster speed (A) Lowest slope = slowest speed (C)

Acceleration

Acceleration is a change in speed or direction.

The distance traveled during a constant time interval

would change.

0m 1m 3m 9m 27m

| | | | |

0s 1s 2s 3s 4s

Change in direction.

Acceleration Formula

- A change in speed per unit of time
- Unit: m/s2 , or m/s/s
- Acceleration = final speed - initial speed
- Change in time
- a = Vf – Vi

∆t

Graphs of Acceleration

- Both graphs show the same motion (acceleration)

Free - Fall

- Motion under the force of gravitation only
- Acceleration due to gravity = 9.8 m/s2

Terminal velocity

- When air resistance upward force equal the downward force of the weight of the object.
- Speed no longer increases but remains constant.

Momentum is Conserved

The total momentum of all objects interacting with one another remains constant regardless of the nature of the forces between the objects.

m1v1i + m2v2i = m1v1f + m2v2f

total initial momentum = total final momentum

!! This relationship is true for all interactions between isolated objects !!

Momentum is conserved in collisions…

Total momentum remains constant for a system of objects that interact with one another. Most conservation-of-momentum problems deal with only two isolated objects. However, when you use conservation of momentum to solve a problem or investigate a situation, it is important to include all objects that are involved In the interaction.

Momentum is conserved for objects pushing away from each other…

Another example of conservation of momentum is when two or more interacting objects that initially have no momentum being moving away from each other. Imagine two skaters pushing away each other . The skaters are both initially at rest with a momentum of p1,i = p2,i = 0. When they push away from each other, they move in opposite directions with equal but opposite momentum so that the total final momentum is also zero.

Newton’s Cradle

This video of Newton’s Cradles displays the conservation of momentum.

Conservation of Momentum Problems

1. A 63.0 kg astronaut is on a spacewalk when the tether line to the shuttle breaks. The astronaut is able to throw a 10.0kg oxygen tank in a direction away from the shuttle with a speed of 12.0 m/sec, propelling the astronaut back to the shuttle. Assuming that the astronaut starts from rest, find the final speed of the astronaut after throwing the tank.

Conservation of Momentum Problems

2. A 76kg boater, initially at rest in a stationary 45kg boat, steps out of the boat and onto the dock. If the boater moves out of the boat with a velocity of 2.5m/sec to the right, what is the final velocity of the boat?

Conservation of Momentum Answers

- 1.90 m/sec
- 2. 4.2 m/sec to the left

Newton’s third law leads to conservation of momentum…

Consider two isolated bumper cars, m1

and m2, before and after they collide.

Before the collision, the velocities of the two bumper cars are v1,I and v2,I. After the collision, their velocities are v1,f and v2,f. The impulse momentum theorem, F1∆t = ∆p, describes the change in momentum of one of the bumper cars. Applied to m1, the impulse momentum theorem gives the following:

F1∆t =m1v1,f - m1v1,I

F2∆t =m2v2,f – m2v2,i

F1 is the force that m2 exerts on m1 during the collision, and F2 is the force that m1 exerts on m2 during the collision. Because the only forces acting in the collision are the forces the two bumper cars exert on each other, Newton’s third law tells us that the force on m2 (F1 = -F2)

1. A horizontal force of 600 N is used to push a box 8 m across a room. Which of these variables must be known to determine the power used in moving the box?

- The weight of the box
- The potential energy of the box
- The time it takes to move the box
- The length of the box

2. The speed of sound in human tissue is about 1600 m/s. If an ultrasound pulse takes 1.5 x 10– 5 s to travel through a tissue, what is the thickness of the tissue?

- 2.4 km
- 2.4 m
- 24 cm
- 24 mm

3. A 0.500 kg ball with a speed of 4.0 m/s strikes a stationary 1.0 kg target. If momentum is conserved, what is the total momentum of the ball and target after the collision?

- 0.0 kgm/s
- 0.5 kgm/s
- 1.0 kgm/s
- 2.0 kgm/s

4. The table contains data for two wrecking balls being used to demolish a building. What is the difference in momentum between the two wrecking balls?

- 300 kgm/s
- 200 kgm/s
- 150 kgm/s
- 0.0 kgm/s

5. A 1-kilogram ball has a kinetic energy of 50 joules. The velocity of the ball is —

- 5 m/s
- 10 m/s
- 25 m/s
- 50 m/s

6. A 0.50 kg ball with a speed of 4.0 m/s strikes a stationary 1.0 kg target. If momentum is conserved, what is the total momentum of the ball and target after the collision?

- 0.0 kgm/s
- 0.5 kgm/s
- 1.0 kgm/s
- 2.0 kgm/s

7. A ball moving at 30 m/s has a momentum of 15 kgm/s. The mass of the ball is —

- 45 kg
- 15 kg
- 2 kg
- 0.5 kg

8. How much work is performed when a 50 kg crate is pushed 15 m with a force of 20 N?

- 300 J
- 750 J
- 1,000 J
- 15,000 J

9. If a force of 100 newtons was exerted on an object and no work was done, the object must have —

- accelerated rapidly
- remained motionless
- decreased its velocity
- gained momentum

10. The weight lifter used a force of 980 N to raise the barbell over her head in 5.21 seconds. Approximately how much work did she do in raising the barbell?

- 380 J
- 982 J
- 2000 J
- 10,000 J

11. A mechanic used a hydraulic lift to raise a 12,054 N car 1.89 m above the floor of a garage. It took 4.75 s to raise the car. What was the power output of the lift?

- 489 W
- 1815 W
- 4796 W
- 30,294 W

12. According to this graph, what was the bicycle’s acceleration between 6 and 10 seconds?

- 0.0 m/s 2
- 0.65 m/s 2
- 1.6 m/s 2
- 6.5 m/s 2

13. An advertisement claims that a certain truck has the most powerful engine in its class. If the engine has more power, which of the following can the truck’s engine do, compared to every other engine in its class?

- Produce fewer emissions
- Operate more efficiently
- Perform work faster
- Accelerate longer

14. What is the net force exerted on a 90.0 kg race-car driver while the race car is accelerating from 0 to 44.7 m/s in 4.50 s?

- 9.8 N
- 20 N
- 201 N
- 894N

15. Starting from rest at the center of a skating rink, two skaters push off from each other over a time period of 1.2 s. What is the force of the push by the smaller skater?

- 16 N
- 32 N
- 88 N
- 100 N

16. A woman lifts a 57-newton weight a distance of 40 centimeters each time she does a particular exercise. It takes her 0.60 second to lift the weight. How much power does she supply for lifting the weight one time?

- 24 W
- 34 W
- 38 W
- 95 W

17. The 500 g cart is moving in a straight line at a constant speed of 2 m/s. Which of the following must the 250 g toy car have in order to maintain the same momentum as the cart?

- An acceleration of 5 m/s2 for 2 seconds
- A potential energy of 20 J
- A constant velocity of 4 m/s
- An applied force of 5 N for 5 seconds

18. The table above shows experimental data collected when four cars moved along a straight-line path. According to these data, which car moved with a constant acceleration of 2 m/s2?

- Car Q
- Car R
- Car S
- Car T

19. A cyclist moves at a constant speed of 5 m/s. If the cyclist does not accelerate during the next 20 seconds, he will travel —

- 0 m
- 4 m
- 50 m
- 100 m

20. An electric toy cart has a mass of 6.0 kilograms and a constant acceleration of 0.50 m/s2. How much work does the net force do on the toy cart as the cart travels 8.0 meters?

- 24 Nm
- 30 Nm
- 40 Nm
- 48 Nm

21. A 2.25 kg fish swims in a pond at a constant rate of 56 meters in 96 seconds. What is the fish’s approximate speed?

- 0.020 m/s
- 0.40 m/s
- 0.58 m/s
- 1.7 m/s

Fireworks

Fireworks displays are often associated with celebrations. Some fireworks are rockets that can be fired into the air, producing colorful patterns of bright light. One rocket design involves a cardboard tube, a propellant, and a fuse. A cap on the tube contains metal salts and explosive powder with a second fuse. The propellant consists of a mixture of carbon (C), sulfur (S), and potassium nitrate (KNO3). Potassium nitrate is a potassium ion (K+) bonded to a nitrate ion (NO3–).

A long cardboard tube is filled with the propellant. When a lit fuse ignites the propellant, the propellant releases oxygen, produces flames, and forces gas out the bottom of the rocket. These actions cause the rocket to rise high into the air.

As the rocket reaches its maximum height, a second fuse ignites an explosion that heats and burns the metal salts. This heating and burning of metal salts produces large colorful flashes. Many people enjoy watching these colorful displays against the night sky.

The use of fireworks can be dangerous. Professionals who use fireworks take many safety precautions while setting up and igniting the displays.

22. Which of the following information would allow the most direct calculation of the average speed of the rocket on its upward flight?

- Thrust force and wind speed
- Maximum height and the time it takes the rocket to reach it
- Rocket mass and the time it takes the rocket to reach the highest point
- Thrust force and the time it takes the rocket to fall to the ground

23. According to the information above, what is the average acceleration of the racer in m/s2? Record and bubble in your answer to the nearest hundredth on the answer document.

0.46

24. The ball in the diagram is moving at a speed of 12 m/s. What is the momentum of the ball in kg m/s? Record and bubble in your answer to the tenths place on the answer document.

2.4

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