A car moving at 10.0 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70.0 kg.
Solution:
Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. In symbols, Newton’s second law of motion is defined to be
Fnet=ΔtΔp
Moreover, Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600.0 m/s in a time of 2.00 ms (milliseconds)?
Solution:
Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:
Δp=FnetΔt
The unknown in this problem is the average force. We can solve the formula for force in terms of the other quantities.
(a) At what speed would a 2.00×104-kg airplane have to fly to have a momentum of 1.60×109 kg⋅m/s (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.
Solution:
Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum p is defined as
p=mv,
where m is the mass of the system and v is its velocity.
Part A.The Speed of an Airplane Given its Momentum
The airplane has the following quantities given:
mairplanepairplane=2.00×104kg=1.60×109kg⋅m/s
Using the formula of momentum, we can solve for the velocity in terms of the other variables. We can then substitute the given quantities to solve for the velocity of the airplane.
Since the momentum of the airplane is 3 orders of magnitude smaller than the ship, the ship will not recoil very much. The recoil would be -0.01 m/s, which is probably not noticeable.
(a) What is the mass of a large ship that has a momentum of 1.60×109 kg⋅m/s, when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s.
Solution:
Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum p is defined as
p=mv,
where m is the mass of the system and v is its velocity.
Part A.The Mass of a Large Ship Given its Momentum
For this part, we are given the following quantities of the ship:
pshipvship=1.60×109kg⋅m/s=48.0km/h=13.3333m/s
From the formula of momentum, we can solve for m in terms of p and v. Then we can substitute the given values to solve for the mass of the ship.
(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7.50 m/s. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s. (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?
Solution:
Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum p is defined as
p=mv,
where m is the mass of the system and v is its velocity.
Part A.The Momentum of the Elephant
We are given the mass and the velocity of the elephant, so we can just directly substitute these values in the formula for momentum.
(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.
Solution:
The translational kinetic energy of an object of mass m moving at speed v is KE=21mv2.
Part A.The Velocity of the Elephant to have the same Kinetic Energy as the Sprinter
First, we need to solve for the kinetic energy of the sprinter.
Part B. How Larger Energies Needed for the Movement of Larger Animals would Relate to Metabolic Rates
If the elephant and the sprinter accelerate to a final velocity of 10.0 m/s, then the elephant would have a much larger kinetic energy than the sprinter. Therefore, the elephant clearly has burned more energy and requires a faster metabolic output to sustain that speed. (Answer)
Suppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a 60.0º slope at constant speed, as shown in Figure 7.34. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by friction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?
Figure 7.34 A rescue sled and victim are lowered down a steep slope.
Solution:
The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle θ between them. In symbols,
W=Fdcosθ
Part A. The Work Done by the Friction on the Sled
First, let us calculate the magnitude of the friction force, Ff. We can do this using the formula,
f=μsN
where f is the friction force, μs is the coefficient of static friction, and N is the normal force directed perpendicular to the surface as shown in the free-body diagram below.
Let us solve for the magnitude of the normal force, N, by summing up forces in the y-direction and equating it to zero, since the body is in equilibrium (moving at constant speed).
Now that we solved the normal force to be 441 newtons, we can now solve for the value of the frictional force, f.
fff=μsN=0.100(441N)=44.1N
We can now substitute this value in the formula for work to solve for the work done by the friction force to the sled. We should also note that the friction force is against the direction of motion making the friction force and the displacement acting in opposite directions. This means that θ=180∘.
Using the same free-body diagram, we can solve for the magnitude of the force on the rope, T. The symbol T is used as this is a tension force from the rope.
Taking the sum of forces in the x-direction and equating it to zero.
This is equivalent to the weight of the sled (and the victim). We can now substitute the weight of the sled and the displacement, knowing that the angle between these two quantities is θ=30∘.
Since the sled moves at a constant speed, the net work done on the sled should be equal to zero. This is validated if we sum up all the works by each individual forces.
A shopper pushes a grocery cart 20.0 m at constant speed on level ground, against a 35.0 N frictional force. He pushes in a direction 25.0º below the horizontal. (a) What is the work done on the cart by friction? (b) What is the work done on the cart by the gravitational force? (c) What is the work done on the cart by the shopper? (d) Find the force the shopper exerts, using energy considerations. (e) What is the total work done on the cart?
Solution:
The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle θ between them. In symbols,
W=Fdcosθ
Part A. The Work Done on the Cart by Friction
In this case, the friction opposes the motion. So, we have the following given values:
F=d=θ=35.0N20.0m180∘
The value of the angle θ indicates that F and d are directed in opposite directions. Substituting these values into the formula,
Part B. Work Done on the Cart by the Gravitational Force
In this case, the gravitational force is directed downward while the displacement is horizontal as shown in the figure below.
We are given the following values:
F=d=θ=mg20.0m90∘
Substituting these values into the work formula, we have
W=W=W=W=Fdcosθ(mg)(20.0m)cos90∘0N⋅m0(Answer)
We can see that the gravitational force does not do any work on the cart because of the angle between the two quantities.
Part C. The Work on the Cart by the Shopper
Since we do not know the force exerted by the shopper, we are going to compute the work done by the shopper on the cart using the Work-Energy Theorem.
The work-energy theorem states that the net work Wnet on a system changes its kinetic energy. That is
Wnet=21mv2−21mv02
Now, we know that the shopper pushes the cart at a constant speed. This indicates that the initial and final velocities are equal to each other, making the net work Wnet is equal to zero.
Wnet=0
We also know that the total work done on the cart is the sum of the work done by the shopper and the friction force.
In this case, the work of the shopper is directed 25 degrees below the horizontal while the displacement is still horizontal. This is depicted in the image below.
We are given the following values:
Wshopper=d=θ=700J20.0m25∘
Substituting these values in the formula for work, we have
How much work is done by the boy pulling his sister 30.0 m in a wagon as shown in Figure 7.33? Assume no friction acts on the wagon.
Figure 7.33 The boy does work on the system of the wagon and the child when he pulls them as shown.
Solution:
The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle θ between them. In symbols,
W=Fdcosθ
In this case, we are given the following values:
Fdθ=50N=30m=30∘
Substituting these values into the equation, we have
You must be logged in to post a comment.