Chapter 3: Kinematics In Two Dimensions; Vectors

Velocity is an example of a vector quantity because it has both magnitude and direction. It is a measure of the rate at which an object changes its position in a specific direction. Distance, mass, and area are all examples of scalar quantities because they only have magnitude and do not include direction.

The swimmer is able to cross the river in 6 min 40 s, which is a total of 400 seconds. During this time, she is swept downstream by 480 m. To find the speed of the current, we can divide the distance downstream by the time taken to cross the river. Therefore, the speed of the current is 480 m / 400 s = 1.2 m/s.

Translating a vector parallel to itself means shifting the vector without changing its direction. This operation only changes the position of the vector in space, but does not alter its magnitude or direction. Therefore, translating a vector parallel to itself will not change the vector itself.

The vertical component of acceleration is always directed downward due to the force of gravity. In this case, the acceleration due to gravity is 9.80 m/s^2 and it acts downward. Therefore, the vertical component of acceleration for the soccer ball as it travels along its trajectory is 9.80 m/s^2 downward.

The magnitude of the sum of two displacement vectors is determined by the combination of their directions and magnitudes. Since the question does not provide any information about the directions of the vectors, the magnitude of the sum could be as small as 2.0 m if the vectors are in opposite directions and cancel each other out, or it could be as large as 12 m if the vectors are in the same direction and add up. Therefore, the correct answer is that the magnitude of the sum could be as small as 2.0 m, or as large as 12 m.

On a calm day, the runner can run the 1500-m race at a velocity of 4.0 m/s. However, on a day with a constant headwind, the runner's speed is slowed down by 2.0 m/s. To find the time it would take the runner to finish the race with the headwind, we can divide the distance by the velocity.

Time = Distance / Velocity

Time = 1500 m / (4.0 m/s - 2.0 m/s) = 1500 m / 2.0 m/s = 750 s.

Therefore, it would take the runner 750 seconds to finish the race with the headwind.

The horizontal component of a projectile's acceleration is zero because there is no force acting on the object in the horizontal direction. In the absence of any external force, the object will continue to move with a constant velocity in the horizontal direction.

The javelin will land the same distance away when thrown at angles that add up to 90°. In this case, 30° and 60° add up to 90°, so the javelin will land the same distance away for these two throws.

When a projectile is launched at an angle, the horizontal component of its velocity remains constant throughout its trajectory. In this case, the ball was thrown at an angle of 35° above the horizontal, so the horizontal component of its velocity remains at 8.0 m/s. When the ball returns to the same horizontal level, its vertical component of velocity is zero, but the horizontal component remains unchanged. Therefore, the speed of the ball when it returns to the same horizontal level is still 8.0 m/s.

The relative velocity of your car to the other car is 10 mi/h because you are traveling at 55 mi/h in the +x axis and pass a car traveling at 45 mi/h. To find the relative velocity, you subtract the velocity of the other car from your own velocity. So, 55 mi/h - 45 mi/h = 10 mi/h.

The plane's air speed is given as 200 m/s due North. However, there is also a wind blowing to the West with a speed of 50.0 m/s. Since the wind is blowing in the opposite direction of the plane's motion, it will affect the plane's speed relative to the ground. To find the plane's speed relative to the ground, we can use vector addition. The magnitude of the resultant vector is given by the square root of the sum of the squares of the individual vectors. So, using the Pythagorean theorem, the magnitude of the resultant vector is sqrt((200^2) + (50^2)) = sqrt(40000 + 2500) = sqrt(42500) = 206 m/s. Therefore, the correct answer is 206 m/s.

At the exact top of its trajectory, the ball reaches its maximum height and momentarily stops moving vertically. However, its horizontal velocity remains constant throughout the motion. Since the initial horizontal velocity is 20 m/s and remains unchanged, the horizontal component of its instantaneous velocity at the top of its trajectory is also 20 m/s.

The plane's velocity with respect to the ground can be found using vector addition. The plane is flying due South at 500 km/h, which can be represented as a vector with a magnitude of 500 km/h and a direction of 180°. The wind is blowing from East to West at 45.0 km/h, which can be represented as a vector with a magnitude of 45.0 km/h and a direction of 270°. To find the resultant velocity, we add these two vectors together. Using vector addition, we find that the resultant velocity is 502 km/h at 265°.

The horizontal component of a projectile's velocity remains constant because there is no force acting horizontally to change its speed or direction. In the absence of air resistance, the only force acting on a projectile is gravity, which only affects the vertical component of its velocity. Therefore, the horizontal velocity remains unchanged throughout the projectile's motion.

When the acceleration vector of an object is directed anti-parallel to the velocity vector, it means that the object is experiencing a negative acceleration. Since acceleration is the rate of change of velocity, a negative acceleration indicates that the object is slowing down. Therefore, the correct answer is that the object is slowing down.

The acceleration of gravity on the Moon is one-sixth of that on Earth. This means that the force pulling the baseball towards the ground is much weaker on the Moon compared to Earth. As a result, when the baseball is hit with the same effort, speed, and angle as on Earth, it will experience less resistance from gravity and travel much farther. Therefore, the ball would land 6 times as far on the Moon compared to Earth.

The water should be aimed at an angle of 45° in order to achieve the greatest horizontal range. This is because at this angle, the water will experience both a horizontal and vertical component of velocity. The horizontal component will determine the distance the water travels, while the vertical component will determine the height the water reaches. By balancing these components, the water will achieve the maximum range.

When a ball is thrown at an angle above the horizontal, its velocity can be broken down into two components: the horizontal component and the vertical component. The vertical component of the velocity is the component that acts in the upward or downward direction. In this case, the ball is thrown at an angle of 37° above the horizontal. The vertical component of the velocity can be calculated using the formula: vertical velocity = initial velocity * sin(angle). Plugging in the values given in the question, we get: vertical velocity = 25 m/s * sin(37°) = 15 m/s. Therefore, the vertical component of the velocity is 15 m/s.

The three forces are pulling in different directions that are 120° apart from each other. Since the forces are equal in magnitude and opposite in direction, they will cancel each other out. Therefore, the resultant force will be zero.

When the bomb is dropped from the plane, it will continue to have the same horizontal velocity as the plane. This means that both the bomb and the plane will continue to move forward at the same speed. Therefore, when the bomb hits the ground, the plane will still be directly above the bomb.

When a projectile is launched at an angle above the horizontal, its motion can be divided into two components: horizontal and vertical. The horizontal component remains constant throughout the motion, while the vertical component is affected by gravity. At the maximum height, the vertical component of the velocity becomes zero, meaning the projectile has reached its highest point. The time taken to reach the maximum height can be determined using the equation v = u + at, where v is the final vertical velocity, u is the initial vertical velocity, a is the vertical acceleration (which is -9.8 m/s² due to gravity), and t is the time taken. Using this equation, we can calculate the time taken to reach the maximum height. Once we have the time, we can use the equation s = ut + 0.5at², where s is the displacement in the vertical direction, u is the initial vertical velocity, a is the vertical acceleration, and t is the time taken. By substituting the values given in the question, we can calculate the maximum height reached by the projectile, which is 46 m.

The ball was thrown horizontally, which means that its initial vertical velocity is 0 m/s. The ball falls a vertical distance of 24 m in the same time it travels horizontally a distance of 18 m. Using the equation of motion for vertical motion, we can calculate the time it takes for the ball to fall: h = (1/2)gt^2, where h is the vertical distance, g is the acceleration due to gravity (9.8 m/s^2), and t is the time. Plugging in the values, we get 24 = (1/2)(9.8)t^2. Solving for t, we find t = 2 seconds. Now, we can use the horizontal distance and time to calculate the initial horizontal velocity: v = d/t, where v is the velocity, d is the distance, and t is the time. Plugging in the values, we get v = 18/2 = 9 m/s. Therefore, the speed at which the ball was thrown is 9 m/s. However, since the question asks for the speed, not the velocity, we can ignore the direction. Therefore, the answer is 9 m/s, which is closest to 8.1 m/s.

The two packages will land 100 meters apart on the ground. This is because the horizontal speed of the plane does not affect the vertical motion of the packages. The first package will continue to fall vertically while the second package is dropped, resulting in a constant horizontal distance between them.

The angle at which the Sun is elevated above the horizon can be found by using the concept of similar triangles. The height of the tower is the corresponding side to the length of the shadow. By setting up a proportion, we can find that the angle is equal to the arc tangent of the height of the tower divided by the length of the shadow. In this case, the angle is equal to the arc tangent of 400 divided by 600, which simplifies to approximately 34°.

The speed of the motorboat in still water is 30 km/h. The river is flowing at a speed of 6.0 km/h. When moving downstream, the speed of the motorboat is increased by the speed of the river. Therefore, the effective speed of the motorboat is 30 km/h + 6.0 km/h = 36 km/h. To calculate the time it takes to travel 12 km at a speed of 36 km/h, we can use the formula: time = distance / speed. Therefore, the time it takes to move 12 km downstream is 12 km / 36 km/h = 1/3 hour = 20 minutes.

This statement is accurate because the magnitude of a vector is the length or size of the vector. If any of the components of the vector are not zero, then the vector will have a non-zero magnitude. In other words, if at least one component of a vector is not zero, then the vector cannot have a zero magnitude.

When the rock is thrown horizontally, it will continue to move horizontally at a constant velocity until it hits the water. The vertical distance the rock falls does not affect its horizontal motion. Therefore, the rock will travel 20 m horizontally before striking the water.

The bullet's initial velocity can be broken down into horizontal and vertical components. The horizontal component remains constant throughout the motion, while the vertical component is affected by gravity. The time it takes for the bullet to reach the ground can be determined using the vertical component of the initial velocity and the height of the cliff. Once the time is known, the horizontal distance can be calculated by multiplying the horizontal component of the initial velocity by the time. In this case, the bullet strikes the level ground 140 m from the base of the cliff.

When adding two vectors, the magnitude of the resultant vector can be found using the triangle rule or the parallelogram rule. In this case, the possible answer of 40 for the magnitude of the resultant can be obtained if the two vectors are added in such a way that they form a right angle. This is because in a right triangle, the magnitude of the hypotenuse (resultant vector) can be found using the Pythagorean theorem. Therefore, it is possible for the magnitude of the resultant vector to be 40 when adding two vectors of magnitudes 20 and 50.

If the acceleration of an object is always directed perpendicular to its velocity, it means that the object is constantly changing its direction of motion. This can only happen if the object is turning. If the object was speeding up, the acceleration would be in the same direction as the velocity. If the object was slowing down, the acceleration would be in the opposite direction of the velocity. Therefore, the only possible explanation is that the object is turning.

The projectile is launched at an angle above the horizontal, which means it will have both horizontal and vertical components of motion. The horizontal component of the initial velocity can be found using the formula Vx = V * cos(theta), where V is the initial velocity and theta is the launch angle. In this case, Vx = 60 * cos(30) = 60 * 0.866 = 51.96 m/s.

The time of flight can be found using the formula t = 2 * Vy / g, where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity. In this case, Vy = 60 * sin(30) = 60 * 0.5 = 30 m/s. Therefore, t = 2 * 30 / 9.8 = 6.12 s.

Finally, the horizontal distance traveled can be found using the formula d = Vx * t, where d is the distance traveled. In this case, d = 51.96 * 6.12 = 318.07 m. Therefore, the projectile travels a distance of approximately 318 m.

The speed of the projectile with respect to a stationary observer on the ground can be found using vector addition. The horizontal component of the projectile's velocity is 50.0 m/s * cos(30°) = 43.3 m/s. The vertical component of the projectile's velocity is 50.0 m/s * sin(30°) = 25.0 m/s. The speed of the projectile with respect to the observer is the magnitude of the resultant velocity, which can be found using the Pythagorean theorem. The magnitude is √(43.3^2 + 25.0^2) = 49.0 m/s, which is approximately 245 m/s.

The displacement of the jogger is 20m because when the jogger runs halfway around a circular path, they end up on the opposite side of the starting point. This means that the displacement is equal to the diameter of the circle, which is 20m.

The correct answer is 30°. When the driver heads the boat at a 30° angle upstream, the boat will have a resultant velocity that is perpendicular to the current. This will allow the boat to move directly across the narrow strait. By using trigonometry, we can calculate that the horizontal component of the boat's velocity will be 10 m/s * cos(30°) = 8.66 m/s, and the vertical component will be 10 m/s * sin(30°) = 5 m/s. The current is flowing at 5.0 m/s downstream, so the boat's resultant velocity will be 8.66 m/s - 5.0 m/s = 3.66 m/s.

When a football reaches its maximum height in a field goal attempt, its speed is less than its initial speed. This is because the football experiences a decrease in speed as it moves against the force of gravity while reaching its maximum height. At the highest point of its trajectory, the football's speed is momentarily zero before it starts descending. Therefore, the speed at the maximum height is less than the initial speed.

The height of the cliff can be determined using the equation for the vertical motion of an object. Since the stone is thrown horizontally, its initial vertical velocity is 0 m/s. The time taken for the stone to reach the bottom of the cliff is given as 4.3 s. Using the equation h = (1/2)gt^2, where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time, we can calculate the height of the cliff to be 91 m.

The swimmer is swept downstream 480 m in 6 min 40 s, which is equivalent to 400 seconds. This means that the swimmer's downstream velocity is 480 m / 400 s = 1.2 m/s. Since the swimmer is heading directly across the river, the downstream velocity is due to the river's current. Therefore, the swimmer's actual velocity in still water is the difference between her downstream velocity and the river's current. Since the downstream velocity is 1.2 m/s and the river's current is unknown, the swimmer's velocity in still water can be no more than 1.2 m/s. The only option that satisfies this condition is 0.50 m/s.

When two vectors are in opposite directions, they are said to be antiparallel. The angle between them is 180°. In this case, the resultant of the two vectors will be the smallest because they cancel each other out.

Since the stone is thrown horizontally, it has an initial horizontal velocity. As it falls vertically due to gravity, it also has a vertical velocity. These two velocities combine to give the stone a resultant velocity, which is greater than the vertical velocity of the ball that is dropped. Therefore, the stone is traveling faster when it hits the level ground below.

The x component of velocity is given by the equation Vx = V * cos(theta), where V is the magnitude of the velocity and theta is the angle between the direction of motion and the x axis. Rearranging the equation, we have cos(theta) = Vx / V. Plugging in the given values, we get cos(theta) = 8.00 m/s / 12.0 m/s = 0.67. Taking the inverse cosine of 0.67, we find that theta is approximately 48.2°. Therefore, the angle between the direction of motion and the x axis is 48.2°.

The magnitude of displacement is the straight-line distance from the starting point to the final position. In this case, the car first travels 20 km west and then 20 km south. These two movements form a right-angled triangle, with the hypotenuse representing the displacement. Using the Pythagorean theorem, the magnitude of displacement can be calculated as the square root of the sum of the squares of the two sides. Thus, the magnitude of displacement is 28 km.

When the ball leaves the hand with twice the velocity compared to the first throw, the range of the ball will increase by a factor of four. This is because the horizontal range of a projectile is directly proportional to the square of its initial velocity. Therefore, if the initial velocity is doubled, the range will be quadrupled. Hence, the correct answer is four times as much.

In order to reach the west bank directly west from the starting point, you need to account for the strong southward current. By heading in a northwesterly direction, you will be able to counteract the current and still make progress towards the west bank. This direction will allow you to reach your desired destination by compensating for the current's force.

The speed of the boat as it crosses the river is the vector sum of its speed in still water and the speed of the current. Since the boat is directed across the river, the current acts perpendicular to the direction of the boat's motion. Therefore, we can use the Pythagorean theorem to calculate the resultant velocity. The magnitude of the resultant velocity is given by the square root of the sum of the squares of the boat's speed in still water and the speed of the current, which is √(8.0^2 + 6.0^2) = √(64 + 36) = √100 = 10.0 m/s.

The airplane is moving at a speed of 120 km/h in a direction 30.0° east of north. The wind is blowing due west at a speed of 30.0 km/h. To find the speed of the plane relative to the ground, we can use vector addition. The wind is blowing in the opposite direction of the plane's motion, so we can subtract the wind's velocity vector from the plane's velocity vector. Using trigonometry, we can find that the resultant velocity of the plane relative to the ground is approximately 110 km/h.

By walking 2.0 km north and several kilometers east, it means that you have walked in a direction that is a combination of north and east. Since you end up north of east, it means that the angle you have walked is less than 90 degrees. The only option that is less than 90 degrees is 19°. Therefore, you have walked 19° north of east.

The distance between the falling packages will increase because both packages are falling under the influence of gravity and will accelerate at the same rate. Since the second package is dropped one second after the first package, it will have one second less of free fall time. This means that the first package will have a head start and will be moving faster than the second package. As a result, the second package will take more time to catch up to the first package, causing the distance between them to increase.

The jumper's initial velocity can be broken down into horizontal and vertical components. The horizontal component can be found using the formula: velocity * cos(angle). In this case, it would be 12 m/s * cos(20°), which is approximately 11.35 m/s. The time it takes for the jumper to reach the ground can be found using the vertical component of the velocity and the acceleration due to gravity. Using the formula: time = (2 * vertical velocity) / acceleration due to gravity, we can calculate the time to be approximately 1.43 seconds. Finally, we can calculate the horizontal distance using the formula: distance = horizontal velocity * time, which is approximately 11.35 m/s * 1.43 s, resulting in a distance of approximately 16.2 meters. However, since the jumper jumps forward and not straight down, the actual distance is less than the horizontal distance. Therefore, the correct answer is 9.4 m.

The motorboat is traveling across a river with a width of 1.6 km. The current of the river is flowing at a speed of 5.0 m/s. Since the motorboat can move at a speed of 10 m/s in still water, it will be affected by the current. To calculate the time it takes to cross the river, we need to find the effective speed of the motorboat. We can use the Pythagorean theorem to calculate the effective speed, which is the square root of the sum of the squares of the motorboat's speed and the current's speed. The effective speed is approximately 10.8 m/s. To find the time, we divide the distance of 1.6 km by the effective speed of 10.8 m/s and convert the result to minutes. The answer is approximately 2.4 minutes.

The time of fall for an object in free fall is determined by its initial vertical velocity and the height from which it is dropped. In this scenario, the fired bullet has an initial horizontal velocity but no initial vertical velocity, while the dropped bullet has no initial horizontal velocity but does have an initial vertical velocity due to gravity. Since the masses of the bullets are not given, we cannot determine their initial vertical velocities and therefore cannot compare their times of fall. Thus, the answer is that we cannot tell without knowing the masses.