Question 1

A comet is observed to pass 1.50 × 10⁸ m from the surface of the Sun. The Sun's radius is 6.96 × 10⁸ m. What is the speed of the comet at this point, if its speed when passing the Earth's orbit, 1.50 × 10¹¹ m from the Sun's center, is 25 km/s? (M_S = 1.989 × 10³⁰ kg.)

Accepted Answer

Using the conservation of energy, the speed of the comet can be calculated. The potential energy and kinetic energy at the Earth's orbit and at the closest approach to the Sun are equated. Solving for the speed at the closest approach gives approximately 559 km/s.

Question 2

A 90.0 kg person is running up some stairs at 0.500 m/s. If the staircase is angled at 45.0° to the horizontal, then the power needed to run up the stairs is

Accepted Answer

The power needed to run up the stairs can be calculated using the formula $P = \frac{W}{t}$, where $W$ is the work done and $t$ is the time. Since the person is moving against gravity, $W = mgh$, where $m$ is the mass, $g$ is the acceleration due to gravity (9.8 m/s$^2$), and $h$ is the height gained. The velocity given is along the stairs, so to find the vertical component (which gives us $h/t$), we use $v_y = v \sin(\theta)$, where $\theta = 45^\circ$. Thus, $v_y = 0.500 \times \sin(45^\circ) = 0.500 \times 0.707 = 0.354$ m/s. The power is then $P = mgv_y = 90.0 \times 9.8 \times 0.354 = 312$ Watts.

Question 3

During a basketball game, a player shoots a ball from half-court. When the ball reaches a height of 4.5 m above the floor, it is moving at 5.0 m/s. Assuming the shot was perfectly made, it will pass through the rim of the basket, 3.05 m above the floor. How fast is the ball moving when it passes through the rim?

Accepted Answer

The answer of During a basketball game, a player shoots...

Question 4

A 1.1 kg seagull takes off from a piling along the California coast and accelerates in 0.50 s to a flight speed of 7.5 m/s. The seagull flies along a slightly inclined path, such that at the end of the 0.50 s period, it is 5.3 m above its point of takeoff. What average power is delivered by the bird's wings during the takeoff?

Accepted Answer

The average power delivered by the bird's wings during takeoff can be calculated by finding the total work done (kinetic plus potential energy change) and dividing it by the time. The kinetic energy change is $\frac{1}{2}mv^2 = \frac{1}{2}(1.1)(7.5)^2$, and the potential energy change is $mgh = (1.1)(9.8)(5.3)$. Adding these and dividing by the time (0.50 s) gives the average power, which is approximately 180 W.

Question 5

When the escape speed from the surface of a spherical object of mass M and radius R is equal to the speed of light, that radius is called the Schwarzschild radius for the given mass. What is the Schwarzschild radius of the Sun, whose mass is 2.0 × 10³⁰ kg? (i.e., if all the Sun's mass were compressed into a sphere of this radius, the escape speed from its surface would be the speed of light-it would become a black hole!)

Accepted Answer

The Schwarzschild radius $ R_s $ is given by the formula $ R_s = \frac{2GM}{c^2} $. For the Sun, with $ G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2 $, $ M = 2.0 \times 10^{30} \, \text{kg} $, and $ c = 3.0 \times 10^8 \, \text{m/s} $, the calculation gives approximately 3.0 km.

Question 6

A blob of clay of mass M is propelled upward from a spring that is initially compressed by an amount d. The spring constant is k. What is the ultimate height h above the unstretched spring's end that the clay will reach?

Accepted Answer

The potential energy stored in the compressed spring is given by: P = 1/2 k d^2 When the clay is released, this potential energy is converted into kinetic energy: P = 1/2 M v^2 where v is the velocity of the clay when it leaves the spring. The kinetic energy is then converted into gravitational potential energy as the clay rises: P = M g h where h is the maximum height the clay reaches above the unstretched spring's end. Equating these three expressions for P, we get: 1/2 k d^2 = 1/2 M v^2 = M g h Solving for h, we get: h = (1/k) (1/2 M v^2) = (1/k) (1/2 k d^2) = 1/2 M d^2/k At maximum height, the velocity of the clay is zero, so we can find v using conservation of energy: 1/2 M v^2 + 1/2 k d^2 = M g h v^2 = 2g(h - d) Substituting this expression for v into the equation for h, we get: h = (1/k) (1/2 M (2g(h - d))) = (1/k) (Mg(h - d)) Solving for h, we get: h = (kd/(2Mg)) + d Therefore, the answer is B, kd²^{/(2Mg) − d}.

Question 7

A 1.1 kg seagull takes off from a piling along the California coast and accelerates in 0.50 s to a flight speed of 7.5 m/s. Assuming the seagull flies along a horizontal path (at the same height as its takeoff), what average power is delivered by the bird's wings during the takeoff?

Accepted Answer

The average power delivered by the bird's wings during takeoff can be calculated using the formula for power, which is $P = \frac{\Delta W}{\Delta t}$, where $\Delta W$ is the work done and $\Delta t$ is the time over which the work is done. The work done to accelerate the bird can also be expressed as the change in kinetic energy, $\Delta KE = \frac{1}{2}mv^2$, where $m$ is the mass of the bird and $v$ is its final velocity. Substituting the given values, we get $\Delta KE = \frac{1}{2}(1.1 \, \text{kg})(7.5 \, \text{m/s})^2 = 30.9375 \, \text{J}$. Since the acceleration takes 0.50 s, the average power is $P = \frac{30.9375 \, \text{J}}{0.50 \, \text{s}} = 61.875 \, \text{W}$, which rounds to 62 W.

Question 8

A blob of clay of mass M is dropped onto a spring from a height h above the end of the spring. If the spring compresses by an amount d as the blob comes to rest, what is the spring constant k?

Accepted Answer

The potential energy (Mgh) and the additional energy due to the spring compression (Mg*d) are converted into spring potential energy (1/2 * k * d^2). Solving for k gives k = 2Mg(h+d)/d^2.

Question 9

An airplane has a mass of 30,000 kg and travels at 300 m/s. If the drag force is 1000 N, then the power the engines have to provide to keep flying at a constant speed is

Accepted Answer

The answer of An airplane has a mass of 30,000...

Question 10

A spring-powered dart gun is unstretched and has a spring constant 10.0 N/m. The spring is compressed by 8.00 cm and a 5.00 gram projectile is placed in the gun. The kinetic energy of the projectile when it is shot from the gun is

Accepted Answer

The kinetic energy of the projectile can be calculated using the formula for the potential energy stored in a compressed spring, which is given by $PE = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the compression distance. Here, $k = 10.0 \, \text{N/m}$ and $x = 8.00 \, \text{cm} = 0.080 \, \text{m}$. Plugging these values into the formula gives $PE = \frac{1}{2} \times 10.0 \, \text{N/m} \times (0.080 \, \text{m})^2 = 0.032 \, \text{J}$. This potential energy is converted into kinetic energy as the projectile is shot from the gun, so the kinetic energy of the projectile is 0.032 J.

Question 11

A 1800 kg car moving at 20 m/s hits an initially uncompressed spring with a spring constant of 2.0 × 10⁶ N/m. The maximum compression of the spring is

Accepted Answer

Before the collision, the car has a kinetic energy of:
KE = (1/2)mv^2 = (1/2)(1800 kg)(20 m/s)^2 = 360,000 J

When the car hits the spring, its kinetic energy is converted into potential energy stored in the compressed spring. At the maximum compression, all of the kinetic energy will be converted into potential energy.

The potential energy stored in a spring is given by:
PE = (1/2)kx^2
where k is the spring constant and x is the displacement (compression) of the spring.

Setting KE equal to PE, we get:
(1/2)mv^2 = (1/2)kx^2
Solving for x, we get:
x = sqrt((mv^2)/k) = sqrt((1800 kg)(20 m/s)^2/(2.0 × 10^16 N/m)) = 0.60 m

Therefore, the maximum compression of the spring is 0.60 m, which corresponds to choice B.

Question 12

A ski lift carries passengers up a 150 m high slope at the rate of one chair every 30.0 seconds. If the chair will hold two passengers with a combined total mass of 190 kg, then the power output of the engine that powers the ski lift is

Accepted Answer

The answer of A ski lift carries passengers up a...

Question 13

A dragster drives a quarter mile (402 m) at a constant rate of acceleration, starting from rest and finishing the race with a final speed of 450 km/hr. The mass of the dragster, together with its driver, is 975 kg. What average power is provided by the engine, if 50% of the power output is delivered to the wheels? Neglect dissipative forces.

Accepted Answer

The average power delivered to the wheels can be calculated using the work-energy principle and then accounting for the efficiency of power delivery. First, convert the final speed to meters per second: $450 \, \text{km/hr} = 450 \times \frac{1000}{3600} = 125 \, \text{m/s}$. The kinetic energy at the end of the quarter mile is $E_k = \frac{1}{2}mv^2 = \frac{1}{2} \times 975 \times (125)^2$. The work done is equal to this kinetic energy, and the power is work done per time. The time $t$ can be found from $v = at$ and $s = \frac{1}{2}at^2$, where $s = 402 \, \text{m}$, $v = 125 \, \text{m/s}$, and $a$ is the acceleration. Solving these equations gives $t$ and then $P = \frac{E_k}{t}$. Since only 50% of the engine's power output is delivered to the wheels, the total power output is twice the power calculated for the wheels. This calculation results in approximately 2.4 MW, considering the steps and the efficiency of power delivery.

Question 14

A student lifts a weight of 10.0 N a distance of 0.5 meters. If this exercise is repeated 50 times in 100 seconds, then the average power output of the student is

Accepted Answer

The work done in one lift is force times distance: 10.0 N * 0.5 m = 5.0 J. Repeating this 50 times results in 250 J of work. Power is work divided by time, so 250 J / 100 s = 2.5 W.

Question 15

A car has a mass of 2,000 kg and travels at 20.0 m/s. If the drag force is 100 N, then the power the engines have to provide to keep moving at constant speed is

Accepted Answer

Power is the rate at which work is done. In this case, the work done is the force applied to the car multiplied by the distance traveled. The force applied is the drag force, which is 100 N. The distance traveled is the speed of the car multiplied by the time it takes to travel that distance. The time it takes to travel that distance is not given, so we cannot calculate the power directly. However, we can use the fact that the car is traveling at a constant speed to determine that the power provided by the engines must be equal to the drag force multiplied by the speed of the car. Therefore, the power provided by the engines is 100 N * 20.0 m/s = 2.0 kW.

Question 16

A 2500 kg car accelerates from rest to a velocity of 10 m/s in 5.0 seconds. The average power of the engine during this acceleration is

Accepted Answer

The answer of A 2500 kg car accelerates from rest...

Question 17

An asteroid of mass 1.4 × 10¹⁴ kg is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its radius?

Accepted Answer

The escape speed $ v_e $ is given by the formula $ v_e = \sqrt{\frac{2GM}{R}} $, where $ G $ is the gravitational constant, $ M $ is the mass of the asteroid, and $ R $ is its radius. Solving for $ R $ using the given escape speed and mass, we find $ R \approx 2.1 $ km.

Question 18

A road bicycle racer rides down the highway from Pike's peak. During one stretch of the ride, he descends 150 m while accelerating from 35 km/hr to some final speed. Assuming friction and air resistance to be negligible, and treating kinetic energy as purely translational (i.e., ignoring any rotational part), what is his final speed?

Accepted Answer

The answer of A road bicycle racer rides down the...

Question 19

A 2000 kg car accelerates from rest to a velocity of 20 m/s in 10 seconds. The average power of the engine during this acceleration is

Accepted Answer

The average power can be calculated using the formula for power, which is the work done over time, $P = \frac{W}{t}$. The work done on the car is equal to the change in kinetic energy, $W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$, where $m$ is the mass, $v$ is the final velocity, and $u$ is the initial velocity. Given that the car accelerates from rest ($u = 0$) to $v = 20$ m/s, the change in kinetic energy is $\Delta KE = \frac{1}{2} \times 2000 \times 20^2 = 400,000$ Joules. The time taken is 10 seconds, so the average power is $P = \frac{400,000}{10} = 40,000$ Watts or 40 kW.

Question 20

An asteroid of radius 2.1 km is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its mass?

Accepted Answer

The answer of An asteroid of radius 2.1 km is...

Quiz 6: Conservation of Energy

A comet is observed to pass 1.50 × 10⁸ m from the surface of the Sun. The Sun's radius is 6.96 × 10⁸ m. What is the speed of the comet at this point, if its speed when passing the Earth's orbit, 1.50 × 10¹¹ m from the Sun's center, is 25 km/s? (M_S = 1.989 × 10³⁰ kg.)

A 90.0 kg person is running up some stairs at 0.500 m/s. If the staircase is angled at 45.0° to the horizontal, then the power needed to run up the stairs is

A blob of clay of mass M is propelled upward from a spring that is initially compressed by an amount d. The spring constant is k. What is the ultimate height h above the unstretched spring's end that the clay will reach?

A 1.1 kg seagull takes off from a piling along the California coast and accelerates in 0.50 s to a flight speed of 7.5 m/s. Assuming the seagull flies along a horizontal path (at the same height as its takeoff) , what average power is delivered by the bird's wings during the takeoff?

A blob of clay of mass M is dropped onto a spring from a height h above the end of the spring. If the spring compresses by an amount d as the blob comes to rest, what is the spring constant k?

An airplane has a mass of 30,000 kg and travels at 300 m/s. If the drag force is 1000 N, then the power the engines have to provide to keep flying at a constant speed is

A spring-powered dart gun is unstretched and has a spring constant 10.0 N/m. The spring is compressed by 8.00 cm and a 5.00 gram projectile is placed in the gun. The kinetic energy of the projectile when it is shot from the gun is

A 1800 kg car moving at 20 m/s hits an initially uncompressed spring with a spring constant of 2.0 × 10⁶ N/m. The maximum compression of the spring is

A ski lift carries passengers up a 150 m high slope at the rate of one chair every 30.0 seconds. If the chair will hold two passengers with a combined total mass of 190 kg, then the power output of the engine that powers the ski lift is

A student lifts a weight of 10.0 N a distance of 0.5 meters. If this exercise is repeated 50 times in 100 seconds, then the average power output of the student is

A car has a mass of 2,000 kg and travels at 20.0 m/s. If the drag force is 100 N, then the power the engines have to provide to keep moving at constant speed is

A 2500 kg car accelerates from rest to a velocity of 10 m/s in 5.0 seconds. The average power of the engine during this acceleration is

An asteroid of mass 1.4 × 10¹⁴ kg is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its radius?

A 2000 kg car accelerates from rest to a velocity of 20 m/s in 10 seconds. The average power of the engine during this acceleration is

An asteroid of radius 2.1 km is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its mass?

Quiz 6: Conservation of Energy

A comet is observed to pass 1.50 × 108 m from the surface of the Sun. The Sun's radius is 6.96 × 108 m. What is the speed of the comet at this point, if its speed when passing the Earth's orbit, 1.50 × 1011 m from the Sun's center, is 25 km/s? (MS = 1.989 × 1030 kg.)

A 90.0 kg person is running up some stairs at 0.500 m/s. If the staircase is angled at 45.0° to the horizontal, then the power needed to run up the stairs is

A blob of clay of mass M is propelled upward from a spring that is initially compressed by an amount d. The spring constant is k. What is the ultimate height h above the unstretched spring's end that the clay will reach?

A 1.1 kg seagull takes off from a piling along the California coast and accelerates in 0.50 s to a flight speed of 7.5 m/s. Assuming the seagull flies along a horizontal path (at the same height as its takeoff) , what average power is delivered by the bird's wings during the takeoff?

A blob of clay of mass M is dropped onto a spring from a height h above the end of the spring. If the spring compresses by an amount d as the blob comes to rest, what is the spring constant k?

An airplane has a mass of 30,000 kg and travels at 300 m/s. If the drag force is 1000 N, then the power the engines have to provide to keep flying at a constant speed is

A spring-powered dart gun is unstretched and has a spring constant 10.0 N/m. The spring is compressed by 8.00 cm and a 5.00 gram projectile is placed in the gun. The kinetic energy of the projectile when it is shot from the gun is

A 1800 kg car moving at 20 m/s hits an initially uncompressed spring with a spring constant of 2.0 × 106 N/m. The maximum compression of the spring is

A ski lift carries passengers up a 150 m high slope at the rate of one chair every 30.0 seconds. If the chair will hold two passengers with a combined total mass of 190 kg, then the power output of the engine that powers the ski lift is

A student lifts a weight of 10.0 N a distance of 0.5 meters. If this exercise is repeated 50 times in 100 seconds, then the average power output of the student is

A car has a mass of 2,000 kg and travels at 20.0 m/s. If the drag force is 100 N, then the power the engines have to provide to keep moving at constant speed is

A 2500 kg car accelerates from rest to a velocity of 10 m/s in 5.0 seconds. The average power of the engine during this acceleration is

An asteroid of mass 1.4 × 1014 kg is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its radius?

A 2000 kg car accelerates from rest to a velocity of 20 m/s in 10 seconds. The average power of the engine during this acceleration is

An asteroid of radius 2.1 km is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its mass?

A comet is observed to pass 1.50 × 10⁸ m from the surface of the Sun. The Sun's radius is 6.96 × 10⁸ m. What is the speed of the comet at this point, if its speed when passing the Earth's orbit, 1.50 × 10¹¹ m from the Sun's center, is 25 km/s? (M_S = 1.989 × 10³⁰ kg.)

A 1800 kg car moving at 20 m/s hits an initially uncompressed spring with a spring constant of 2.0 × 10⁶ N/m. The maximum compression of the spring is

An asteroid of mass 1.4 × 10¹⁴ kg is observed to have an escape speed equal to a typical person's maximum jumping speed, 3.0 m/s. What is its radius?