kinematics
Which of the following graphs best shows the velocity versus time of an object originally moving upward in the presence of air friction?
(A) Constant positive velocity
(B) Exponential decay from positive to zero
(C) Linear decrease from positive to negative
(D) Curve starting positive, crossing zero, and asymptoting to a negative constant value
(E) Curve starting positive, crossing zero, then rising back to zero
(E) Curve starting positive, crossing zero, then rising back to zero
A packing crate with mass m= 115 kg is slid up a 5.00 m long ramp which makes an angle of 20.0 degrees with respect to the horizontal by an applied force of F = 1.00 ×103 N directed parallel to the ramp’s incline.
A frictional force of magnitude f = 4.00 ×102 N resists the motion. If the crate starts from rest, what is its speed at the top of the ramp?
(A) 4.24 m/s
(B) 5.11 m/s
(C) 7.22 m/s
(D) 8.26 m/s
(E) 9.33 m/s
(A) 4.24 m/s
A spring stretched to double its unstretched length has a potential energy U0. If the spring is cut in half, and each half spring is stretched to double its unstretched length, then the total potential energy stored in the two half springs will be
(A) 4U0
(B) 2U0
(C) U0
(D) U0/2
(E) U0/4
(C) U0
A light, uniform, ideal spring is fixed at one end. If a mass is attached to the other end, the system oscillates with angular frequency ω. Now suppose the spring is fixed at the other end, then cut in half. The mass is attached between the two half springs. The new angular frequency of oscillations is
(A) ω/2
(B) ω
(C) √2ω
(D) 2ω
(E) 4ω
(d) 2w
A mint produces 100,000 coins. Upon weighing some of them with a precise scale, officials find that the coins vary slightly in weight, with an independent uncertainty of 1%. About how many coins must be randomly sampled and weighed in order to determine the total weight of the coins to within 0.1% uncertainty? Assume there are no sources of systematic uncertainty.
(A) Almost all of the coins must be weighed.
(B) 10,000
(C) 1,000
(D) 100
(E) 10
(D) 100
A car has a maximum acceleration of a0 and a minimum acceleration of −a0. The shortest possible time for the car to begin at rest, then arrive at rest at a point a distance d away is:
(A) sqrt(d/2a0)
(B) sqrt(d/a0)
(C) sqrt(2d/a0)
(D) sqrt(3d/a0)
(E) 2sqrt(d/a0)
(E) 2sqrt(d/a0)
A massless cable of diameter 2.54 cm (1 inch) is tied horizontally between two trees 18.0 m apart. A tightrope walker stands at the center of the cable, giving it a tension of 7300 N. The cable stretches and makes an angle of 1.50 deg with the horizontal.
The Young’s modulus is defined as the ratio of stress to strain, where stress is the force applied per unit area and strain is the fractional change in length ∆L/L. The cable’s Young’s modulus is
(A) 1.5 ×106 N/m2
(B) 2.0 ×108 N/m2
(C) 2.2 ×109 N/m2
(D) 2.4 ×1010 N/m2
(E) 4.2 ×1010 N/m2
(E) 4.2 ×1010 N/m2
A mass of 3M moving at a speed v collides with a mass of M moving directly toward it, also with a speed v. If the collision is completely elastic, the total kinetic energy after the collision is Ke. If the two masses stick together, the total kinetic energy after the collision is Ks. What is the ratio Ke/Ks?
(A) 1/2
(B) 1
(C) √2
(D) 2
(E) 4
(E) 4
A satellite is in a circular orbit about the Earth. Over a long period of time, the effects of air resistance decrease the satellite’s total energy by 1 J. The kinetic energy of the satellite
(A) increases by 1 J.
(B) remains unchanged.
(C) decreases by 1/2 J.
(D) decreases by 1 J.
(E) decreases by 2 J.
(A) increases by 1 J.
Raindrops with a number density of n drops per cubic meter and radius r0 hit the ground with a speed v0. The resulting pressure on the ground from the rain is P0. If the number density is doubled, the drop radius is halved, and the speed is halved, the new pressure will be
(A) P0
(B) P0/2
(C) P0/4
(D) P0/8
(E) P0/16
(E) P0/16
A ball is launched straight toward the ground from height h. When it bounces off the ground, it loses half of its kinetic energy. It reaches a maximum height of 2h before falling back to the ground again. What was the initial speed of the ball?
(A) sqrt(gh)
(B) sqrt(2gh)
(C) sqrt(3gh)
(D) sqrt(4gh)
(E) sqrt(6gh)
(E) sqrt(6gh)
A coin of mass m is dropped straight down from the top of a very tall building. As the coin approaches terminal speed, which is true of the net force on the coin?
(A) The net force on the coin is upward.
(B) The net force on the coin is 0.
(C) The net force on the coin is downward, with a magnitude less than mg.
(D) The net force on the coin is downward, with a magnitude equal to mg.
(E) The net force on the coin is downward, with a magnitude greater than mg.
(C) The net force on the coin is downward, with a magnitude less than mg.
A car is driving against the wind at a constant speed v0 relative to the ground. The wind direction is always opposite to the car’s velocity, but its speed fluctuates about an average speed of v relative to the ground. The air drag force is Avrel2, where A is a constant and vrel is the relative speed between the car and the wind. What is the average rate P of energy dissipation due to the air resistance?
(A) P= Av0(v0 + v)2
(B) P >Av0(v0 + v)2
(C) P <Av0(v0 + v)2
(D) Both (B) and (C) are possible depending on how v fluctuates.
(E) Both (A) and (C) are possible depending on how v fluctuates.
(B) P >Av0(v0 + v)2
A spherical cloud of dust has uniform mass density ρ and radius R. Satellite A of negligible mass is orbiting the cloud at its edge, in a circular orbit of radius R, and satellite B is orbiting the cloud just inside the cloud, in a circular orbit of radius r, with r <R. If vi is the speed of satellite i and Ti is the period of satellite i, which of the following is true? Neglect any drag forces from the dust.
(A) TA >TB and vA >vB
(B) TA >TB and vA <vB
(C) TA <TB and vA >vB
(D) TA <TB and vA <vB
(E) TA = TB and vA >vB
(E) TA = TB and vA >vB
To test the speed of a model car, you time the car with a stopwatch as it travels a distance of 100 m.
You record a time of 5.0 s, and your measurement has an uncertainty of 0.2 s. What is the uncertainty in your estimate of the car’s speed? Assume that the car travels at a constant speed and the distance of 100 m is known very precisely.
(A) v= 20 ±0.16 m/s
(B) v= 20 ±0.8 m/s
(C) v= 20 ±1.0 m/s
(D) v= 20 ±1.25 m/s
(E) v= 20 ±4.0 m/s
(B) v= 20 ±0.8 m/s
A train starts from city A and stops in city B. The distance between the cities is s. The train’s maximal acceleration is a1 and its maximal deceleration is a2. What is the shortest time in which the train can travel between A and B?
(A) 2sqrt(s/(a1+a2))
(B) 2sqrt(s/a1a2)
(C) sqrt(2s(a1+a2)/(a1a2))
(D) sqrt(2sa2/(a1(a1+a2)))
(E) sqrt(s*sqrt(a1a2)/(a1+a2)2)
(C) sqrt(2s(a1+a2)/(a1a2))
The maximal tension per area a material can sustain without failure is called its tensile strength. Plain steel has a tensile strength of 415 MPa. What is the maximal mass one can hang on a vertical steel rod of negligible mass and a diameter of 2 cm?
(A) 1300 kg
(B) 5200 kg
(C) 13 000 kg
(D) 52 000 kg
(E) The answer depends on the length of the steel rod.
(C) 13 000 kg
A collision occurs between two masses. In each inertial reference frame, one can compute the change in total momentum ∆P and the change in total kinetic energy ∆K due to the collision. Which of the following is true?
(A) ∆P and ∆K do not depend on the frame.
(B) ∆P and ∆K do not depend on the frame for perfectly elastic collisions, but ∆P may depend on the frame for inelastic collisions.
(C) ∆P and ∆K do not depend on the frame for perfectly elastic collisions, but ∆K may depend on the frame for inelastic collisions.
(D) ∆P and ∆K do not depend on the frame for perfectly elastic collisions, but both may depend on the frame for inelastic collisions.
(E) ∆P and ∆K may both depend on the frame, for both perfectly elastic and inelastic collisions.
(A) ∆P and ∆K do not depend on the frame.
A ping-pong ball (a hollow spherical shell) with mass m is placed on the ground with initial velocity v0 and zero angular velocity at time t = 0. The coefficient of friction between the ping-pong ball and the ground is µs = µk = µ. The time the ping-pong ball begins to roll without slipping is
(A) t= (2/5)v0/µg
(B) t= (2/3)v0/µg
(C) t= v0/µg
(D) t= (5/3)v0/µg
(E) t= (3/2)v0/µg
(A) t= (2/5)v0/µg
A juggler juggles N identical balls, catching and tossing one ball at a time. Assuming that the juggler requires a minimum time T between ball tosses, the minimum possible power required for the juggler to continue juggling is proportional to
(A) N0
(B) N1
(C) N2
(D) N3
(E) N4
(C) N2
A ball is released from rest above an inclined plane and bounces elastically down the plane. As the ball progresses down the plane, the time and the distance between each collision will:
(A) remain the same, and increase.
(B) increase, and remain the same.
(C) decrease, and increase.
(D) decrease, and remain the same.
(E) both remain the same.
(A) remain the same, and increase.
Paul the Giant stands outside on a force-meter calibrated in Newtons, which reads 5000 N. Paul is wearing a large cowboy hat, which has horizontal cross-sectional area A= 1 m2 and completely covers both him and the scale when seen from directly above. At time t = 0, rain begins to fall vertically downward on Paul, and any rain that hits his hat is collected in the hat’s brim. The raindrops have a constant downward speed of 1 m/s, and the rain accumulates on the ground at a rate of 1 mm/s. What is the reading (in N) on the scale as a function of the time t>0 (in s)? The density of water is 1000 kg/m3.
(A) 5001 + 11t
(B) 5001 + 10t
(C) 5000 + 11t
(D) 5001 + 1.1t
(E) 5001 + t
(B) 5001 + 10t
A solid sphere sits at the top of a ramp of height h inclined at angle θ to the horizontal. Both the static and kinetic coefficients of friction between the sphere and incline are µk = µs = 0.2. The sphere is released from rest at the top of the incline. For which of the following values of θ is the total translational plus rotational kinetic energy of the sphere greatest when it reaches the bottom of the incline?
(A) 10◦
(B) 45◦
(C) 60◦
(D) 80◦
(E) The mechanical energy is the same for all choices.
(A) 10◦
A mass is attached to one end of a rigid rod, while the other end of the rod is attached to a fixed horizontal axle. Initially the mass hangs at the end of the rod and the rod is vertical. The mass is given an initial kinetic energy K. If K is very small, the mass behaves like a pendulum, performing small-angle oscillations with period T0. As K is increased, the period of the motion for the mass
(A) remains the same.
(B) increases, approaching a finite constant.
(C) decreases, approaching a finite non-zero constant.
(D) decreases, approaching zero.
(E) initially increases, then decreases.
(E) initially increases, then decreases.
The depth of a well, d, is measured by dropping a stone into it and measuring the time tuntil the splash is heard at the bottom. What is the smallest value of d for which ignoring the time for the sound to travel gives less than a 5% error in the depth measurement? The speed of sound in air is 330 m/s.
(A) 3.5 m
(B) 7 m
(C) 14 m
(D) 54 m
(E) 330 m
(C) 14 m