Chapter 10: Projectile And Satellite Motion

An Earth satellite is considered to be freely falling around the Earth because it is constantly under the influence of Earth's gravitational pull. The satellite is in a state of continuous free fall, where the gravitational force is balanced by its forward motion, resulting in a stable orbit around the Earth. This allows the satellite to remain in space without any propulsion or external forces.

When the woman is on the surface of the Earth, she experiences the force of gravity, which gives her a weight of 490 newtons. However, if she were floating freely inside a space habitat far away from Earth, she would be outside the influence of Earth's gravity and therefore would not experience any weight. However, her mass would remain the same at 50 kilograms, as mass is a measure of the amount of matter an object contains and is independent of gravity. Therefore, the correct answer is that she would have less weight and the same mass.

Satellites in orbit closer to the Earth experience stronger gravitational pull, which causes them to move faster and complete their orbits more quickly. In contrast, satellites in orbit farther from the Earth experience weaker gravitational pull, resulting in slower orbital speeds and longer periods to complete their orbits. Therefore, the period of satellites in orbit far from the Earth is longer.

When a bullet is fired horizontally from a rifle, it already has an initial vertical velocity component due to the force of gravity acting on it. As a result, it begins to fall as soon as it leaves the barrel, regardless of whether air friction affects its speed or not. This is because the force of gravity constantly pulls the bullet downwards, causing it to follow a curved trajectory.

When a projectile is launched vertically upward, it experiences a constant acceleration due to gravity, causing it to slow down until it reaches its highest point. At this point, its velocity becomes zero, and it starts to fall back down. As there is no air resistance, the projectile will regain its initial velocity of 50 m/s when it returns to its starting point. Therefore, the speed upon returning to its starting point is 50 m/s.

The correct answer is their distances from the sun. The speeds of the planets around the sun are determined by their distances from the sun. According to Kepler's laws of planetary motion, planets closer to the sun have shorter orbital periods and higher speeds, while planets farther from the sun have longer orbital periods and lower speeds. This is because the gravitational force between the sun and the planets decreases with distance, resulting in slower speeds for planets that are farther away.

When a projectile is fired vertically from the surface of the Earth at a speed of 5 km/s, it will rise against the force of gravity until it reaches its maximum height. At this point, the projectile's vertical velocity will become zero and it will start falling back to the Earth's surface due to the gravitational pull. Since there is no horizontal velocity given, the projectile will not enter into a circular or elliptical orbit around the Earth. Therefore, the correct answer is that the projectile will rise and fall back to the Earth's surface.

When a bowling ball is rolled off the edge of a table, it experiences only vertical acceleration due to gravity, while its horizontal velocity remains unchanged. This is because there are no horizontal forces acting on the ball, such as air resistance or a push. Therefore, the horizontal component of velocity remains constant throughout the fall.

Earth satellites are typically more than 100 km high to be above the Earth's atmosphere. This is because the atmosphere can interfere with satellite signals and cause disruptions in communication and data transmission. By placing satellites above the atmosphere, they can operate more efficiently and effectively without any interference. Additionally, being above the atmosphere also allows satellites to have a better view of Earth's surface for various purposes such as weather monitoring, navigation, and remote sensing.

The radial velocity of an Earth satellite refers to its velocity perpendicular to the surface of the Earth. This means that it is the component of the satellite's velocity that is directed towards or away from the center of the Earth. The other options are incorrect because they do not accurately describe the direction of the satellite's velocity.

If the moon were farther from the Earth, it would take longer for the moon to complete its orbit around the Earth. This means that the time it takes for the moon to go through its phases, which is roughly a month, would also be longer. Therefore, the lunar month would be more than 28 days if the moon were farther from the Earth.

If the moon were closer to the Earth, the gravitational pull between them would be stronger. This increased gravitational force would cause the moon to orbit the Earth at a faster rate, resulting in a shorter lunar month. Therefore, the lunar month would be less than 28 days if the moon were closer to the Earth.

When a stone is thrown horizontally, its initial vertical velocity is zero. Therefore, the stone only experiences the force of gravity acting vertically downwards. In one second, the stone will fall a distance of 5 meters due to the acceleration of gravity (9.8 m/s^2) acting on it. Thus, its vertical distance below the top of the cliff after one second is 5 meters.

The bullet that is fired at an angle of 2 degrees below the horizontal will hit the ground first. This is because the horizontal component of its velocity will contribute to its forward motion, while the vertical component will contribute to its downward motion. On the other hand, the bullet that is dropped from the level of the rifle will only be affected by gravity pulling it straight down. Therefore, the fired bullet will reach the ground before the dropped bullet.

An Earth satellite in an elliptical orbit experiences varying distances from the Earth. According to Kepler's laws of planetary motion, a satellite moves faster when it is closer to the body it orbits. Therefore, when the satellite is nearest to the Earth in its elliptical orbit, it travels at its fastest speed.

The speed of a satellite is primarily determined by its distance from the planet and its mass. The presence of a large ocean, island, mountain range, plain, or plateau does not significantly affect these factors. Therefore, none of the above choices would have a significant impact on the satellite's speed.

The correct answer is the planet nearest the sun. This is because the closer a planet is to the sun, the stronger the gravitational pull it experiences. This gravitational pull causes the planet to move faster in its orbit around the sun. Therefore, the planet nearest the sun will have the highest orbital speed and will be the fastest moving planet in the solar system.

The bullet will be 100 meters downrange after 1 second because the horizontal velocity of the bullet remains constant throughout its trajectory. Therefore, in 1 second, the bullet will travel 100 meters horizontally.

The minimal orbital speed about Jupiter would be more than 8 km/s because Jupiter is a much larger and more massive planet compared to Earth. The gravitational pull of Jupiter is stronger, requiring a higher speed to maintain a stable orbit.

When an object is thrown upward at a 45-degree angle, it experiences the force of gravity pulling it downward. This force causes the object to deviate from its straight-line path. After 1 second, the object will have fallen a certain distance below the straight line due to gravity. Since the question states that the object is about 5 m below the straight line after 1 second, this suggests that gravity has caused the object to fall downward during that time period.

The dropped bullet will hit the ground first because it is not affected by the horizontal component of its velocity. The fired bullet, on the other hand, has both a horizontal and vertical component of velocity. Even though the fired bullet is initially moving faster due to the force of the rifle, it will take longer to reach the ground because it also has to cover horizontal distance before hitting the ground. Therefore, the dropped bullet will hit the ground first.

The tangential velocity of an Earth satellite is its velocity parallel to the surface of the Earth because satellites in orbit around the Earth move in a circular or elliptical path. This means that their velocity is always perpendicular to the radius of their orbit, which is directed towards the center of the Earth. Since the radius is perpendicular to the surface of the Earth, the tangential velocity must be parallel to the surface.

According to Kepler's laws, the paths of planets about the sun are ellipses. Kepler's first law, also known as the law of orbits, states that planets move in elliptical orbits with the sun at one of the focal points. This means that the path of a planet is not a circle, parabola, or a straight line, but rather an ellipse. Therefore, the correct answer is ellipses.

Pluto takes a longer time to orbit the sun than the Earth because it has much further to go. Additionally, Pluto also goes much slower in its orbit compared to Earth. Both of these factors contribute to the longer orbital period of Pluto.

The projectile will rise and fall back to the Earth's surface because its initial velocity is not sufficient to achieve escape velocity from Earth's gravitational pull. Therefore, it will follow a parabolic trajectory and eventually return to the surface.

If a projectile is launched at an angle of 15 degrees above the horizontal and lands down range, another projection angle of 75 degrees would produce the same down-range distance. This is because the range of a projectile depends on the horizontal component of its velocity. The maximum range is achieved when the projectile is launched at a 45-degree angle. As the angle increases beyond 45 degrees, the range decreases. Therefore, if the initial angle is 15 degrees, the same down-range distance can be achieved by increasing the angle to 75 degrees.

The boat is crossing the river at a speed of 1 m/s due east, while the river is flowing south at a speed of 1 m/s. This means that the boat is effectively moving at a diagonal speed of √(1^2 + 1^2) = √2 m/s. Since the river is 100 m wide, it will take the boat 100/√2 ≈ 70.71 seconds to cross the river. During this time, the boat will also be carried south by the river's current at a speed of 1 m/s. Therefore, the boat will end up 70.71 m farther south than where it started, resulting in the correct answer of 100 m farther south than where it started.

If a bullet is fired horizontally over level ground, it will have a constant horizontal velocity. This means that the time it takes for the bullet to hit the ground is solely determined by the vertical motion. The vertical motion is influenced by the force of gravity, which causes the bullet to accelerate downward. Since the bullet is fired horizontally, the initial vertical velocity is zero. Therefore, the time it takes for the bullet to hit the ground is the same regardless of its initial speed. This is why the bullet would still hit the ground in 0.5 seconds even if it had been fired with twice the speed.

The correct answer is perigee. The term "apogee" refers to the point in a satellite's orbit that is farthest from the Earth, while "perigee" refers to the point that is closest to the Earth. When a satellite is at the perigee, it is closer to the Earth's gravitational pull, which results in a greater acceleration. Therefore, the acceleration is greater for a satellite when it is at the perigee.

A satellite near the Earth takes about an hour and a half to make a full circle. The moon is located much farther away from the Earth compared to the satellite. Therefore, it would take a longer time for a satellite located as far away as the moon to orbit the Earth. The moon takes approximately 28 days to complete one orbit around the Earth.

When an object is dropped and freely falls to the ground, its acceleration is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2 or 1 g. When the object is thrown upward at an angle, it still experiences the acceleration due to gravity acting downward. This is because gravity always acts vertically downward regardless of the object's motion. Therefore, the acceleration of the object when thrown upward at an angle will still be 1 g, but in the downward direction.

If a satellite's radial velocity is zero at all times, it means that its distance from the center of the orbit remains constant. This can only happen in a circular orbit, where the gravitational force acting on the satellite is balanced by its centripetal force. In a parabolic or elliptical orbit, the satellite's distance from the center would vary, resulting in a non-zero radial velocity. Geosynchronous orbit refers to a specific type of circular orbit where the satellite's period of rotation matches the Earth's rotation, but it does not necessarily imply zero radial velocity.

When a rock is thrown upward at an angle of 50 degrees with respect to the horizontal, the only force acting on it is gravity. Gravity only affects the vertical component of velocity, causing the rock to slow down as it rises and speed up as it falls. However, the horizontal component of velocity is not affected by gravity. Therefore, neglecting air drag, the horizontal component of velocity remains unchanged throughout the motion of the rock.

The correct answer is "it is not acted on by any horizontal forces." This is because in the absence of air resistance, there are no external forces acting horizontally on the projectile. Therefore, the projectile will continue to move horizontally with a constant velocity. The other options are incorrect because they do not fully explain why the projectile maintains its horizontal component of velocity.

The tangential velocities of the planets prevent them from crashing into the sun. This is because the planets are moving in a circular orbit around the sun, and the tangential velocity provides the necessary centrifugal force to balance the gravitational force between the planet and the sun. Without the tangential velocities, the planets would be pulled directly towards the sun and collide with it.

Both Newton and Kepler considered forces on the planets. Kepler hypothesized that the direction of the forces acting on the planets was along their directions of travel. This means that the force exerted on a planet would be in the same direction as its motion. Kepler's hypothesis was based on his observations of the planets' orbits and his belief that there must be a force causing them to move in a curved path. This hypothesis was later proven incorrect by Newton, who proposed that the force acting on the planets was actually directed toward the sun.

When one satellite appears to overtake another from Earth, it means that the faster satellite is moving ahead of the slower one in their respective orbits. Since the question asks for the position of the faster satellite, the correct answer is "lower." This means that the faster satellite is positioned at a lower altitude or closer to the Earth's surface compared to the slower satellite.

The airplane is traveling towards the northeast, which is a diagonal direction. To find the component of velocity due north, we need to consider the northward direction. Since the northeast direction is at a 45-degree angle with the north, we can use trigonometry to find the northward component of velocity. The northward component is equal to the velocity multiplied by the cosine of the angle. In this case, the northward component is 141 km/h multiplied by the cosine of 45 degrees, which is approximately 100 km/h.

Speed is not a vector quantity because it only has magnitude and does not have a specific direction. Velocity and acceleration, on the other hand, are vector quantities as they have both magnitude and direction. Therefore, the correct answer is speed.

When an airplane is flying in a crosswind, the wind affects its ground speed. In this scenario, the airplane is flying at 100 km/h, and there is a 100 km/h hurricane crosswind. The crosswind will not affect the airplane's speed directly, but it will cause the airplane to drift sideways. The airplane's ground speed is the vector sum of its airspeed and the crosswind. Since the crosswind is equal to the airplane's airspeed, the ground speed will be the square root of the sum of the squares of the airspeed and crosswind. Using the Pythagorean theorem, the ground speed is calculated to be approximately 141 km/h.

The period of a satellite, the time it takes for a complete revolution, depends on the satellite's radial distance. This is because the gravitational force between the satellite and the object it is orbiting is directly proportional to the square of the distance between them. As the distance increases, the gravitational force decreases, resulting in a longer period for the satellite's revolution. Therefore, the radial distance plays a crucial role in determining the period of a satellite.

The boat needs to travel directly east to land at a point due east of its starting point. However, the river is flowing due south, which means that the boat will be carried downstream while trying to travel east. Since the boat can only move at a speed of 1 m/s relative to the water, it will not be able to counteract the river's flow and reach its desired destination. Therefore, the boat cannot be pointed in any direction to successfully land due east.

When a projectile is fired into the air at an angle of 50 degrees, it follows a curved path known as a parabola. The range of the projectile is determined by the initial velocity and the angle of projection. Since the target is at the same level as the projectile's starting point, the range remains the same. In order for the projectile to land on the target, it must have the same range. Among the given options, firing the projectile at an angle of 40 degrees will result in the same range, allowing it to land on the target.

Speeding up the orbital speed of the Earth so it escapes the sun would require greater energy than slowing down the orbital speed of the Earth so it crashes into the sun. This is because in order to escape the sun's gravitational pull, the Earth would need to increase its velocity significantly, requiring a substantial amount of energy. On the other hand, slowing down the Earth's orbital speed would only require reducing its velocity, which would require less energy compared to escaping the sun's gravitational pull.

The moon travels the fastest when it is closest to the Earth because of the gravitational pull between the two bodies. As the moon gets closer to the Earth, the gravitational force increases, causing the moon to accelerate and move faster in its orbit.

The rocket should fire off at the perigee, where it is closest and fastest. This is because at the perigee, the rocket is at its lowest point in the elliptical orbit and therefore has the highest velocity. By firing off at this point, the rocket can take advantage of its already high velocity and require less additional fuel to reach escape velocity. Firing off at the apogee, where the rocket is farthest and slowest, would require more fuel to overcome the lower velocity and reach escape velocity.

The question states that the minimal orbital speed around the Earth is about 8 km/s. Since the moon has less mass than the Earth, its gravitational pull is weaker. Therefore, an object would require less speed to maintain a stable orbit around the moon compared to the Earth. Hence, the minimal orbital speed around the moon would be less than 8 km/s.

The projectile fired at 30 degrees will hit the ground first because it has a smaller angle of launch, which means it has a greater horizontal component of velocity. The horizontal component of velocity determines how fast the projectile moves horizontally, while the vertical component determines how high the projectile goes. Since both projectiles are fired at the same speed, the one with a smaller angle will have a greater horizontal velocity and therefore reach the ground first.

When a rock is thrown upward at an angle of 50 degrees with respect to the horizontal, its initial velocity is divided into two components - horizontal and vertical. The vertical component of velocity determines the rock's upward or downward motion. As the rock rises, the force of gravity acts against its upward motion, causing its vertical velocity to decrease. Therefore, the correct answer is that the vertical component of velocity decreases.

When the rock reaches the top of its path, its velocity becomes zero, but its acceleration remains the same. This is because acceleration is the rate of change of velocity, and even though the velocity is momentarily zero, the acceleration is still acting on the rock due to the force of gravity. As the rock falls a short distance from the top, its acceleration continues to be the same, as there is no change in the force acting on it. Therefore, the acceleration of the rock is the same as it was at the top of its path.