Chapter 2: Describing Motion: Kinematics In One Dimension

The car is traveling at a speed of 90 km/h. To find the time it takes for the car to travel 400 km, we divide the distance by the speed. 400 km divided by 90 km/h equals approximately 4.4 hours. Therefore, it takes the car 4.4 hours to travel 400 km.

The explanation for the correct answer "remains constant" is that both balls are dropped from the same building, which means they experience the same acceleration due to gravity. This acceleration remains constant throughout the motion, regardless of the time at which the balls were dropped. Therefore, the difference in their speeds will remain constant as well.

The airplane increases its speed from 100 m/s to 160 m/s, a difference of 60 m/s. The average rate of increase is given as 15 m/s^2. Using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can rearrange the formula to solve for t. Plugging in the values, we have 160 = 100 + 15t. Solving for t, we get t = 4.0 seconds. Therefore, it takes 4.0 seconds for the complete increase in speed.

The time it takes for an object to reach its maximum height when thrown upwards can be calculated using the formula: t = v/g, where v is the initial velocity and g is the acceleration due to gravity. In this case, the initial velocity is 12 m/s and the acceleration due to gravity on planet X is 1.5 m/s^2. Plugging these values into the formula, we get t = 12 m/s / 1.5 m/s^2 = 8.0 s. Therefore, it takes 8.0 seconds for the object to reach its maximum height.

The average velocity of an object is equal to the instantaneous velocity only when the velocity is constant. This means that the object is moving at a consistent speed and in a straight line without any changes in direction. In such cases, the average velocity over a given time interval will be the same as the velocity at any specific moment within that interval. However, if the velocity is changing, either in magnitude or direction, the average velocity will not be equal to the instantaneous velocity.

To convert miles per hour (mi/h) to meters per second (m/s), we need to multiply the given value by a conversion factor. The conversion factor is 1609 m/1 mi since 1 mile is equal to 1609 meters. By multiplying 55 mi/h by the conversion factor, we get 55 mi/h * 1609 m/1 mi = 88,495 m/h. To convert this to m/s, we need to divide by 3600 (since there are 3600 seconds in an hour). Therefore, 88,495 m/h / 3600 s/h = 24.58 m/s, which can be rounded to 25 m/s.

The given information states that the object is experiencing an acceleration of 2.0 m/s^2. Acceleration is the rate of change of velocity, so if the object is experiencing an acceleration of 2.0 m/s^2, it means its velocity is increasing by 2.0 m/s every second. Therefore, the correct answer is "increasing its velocity by 2.0 m/s in every second."

The average acceleration of an object can be calculated using the formula: average acceleration = (final velocity - initial velocity) / time. However, in this question, the time is not given. Instead, the distance is provided. To solve this, we can use the formula: average acceleration = (final velocity^2 - initial velocity^2) / (2 * distance). Plugging in the values, we get (80^2 - 40^2) / (2 * 200) = 12 m/s^2. Therefore, the correct answer is 12 m/s^2.

When the ball reaches the highest point, its velocity is zero because it momentarily stops moving before it starts falling back down. However, its acceleration is not zero because gravity is still acting on the ball, pulling it downwards. The acceleration due to gravity is constant and always acts in the downward direction, even when the ball is at its highest point.

As the ball is going up, its velocity points upward because it is moving in the opposite direction of gravity. However, its acceleration points downward because gravity is constantly pulling it downward, causing it to slow down and eventually reverse direction. Therefore, the correct answer is that its velocity points upward and its acceleration points downward.

The slope of a position versus time graph represents the rate at which an object's position is changing over time. This rate is known as velocity, which is the speed and direction of an object's motion. Therefore, the correct answer is velocity.

The slope of a velocity versus time graph gives the rate of change of velocity, which is acceleration. This is because acceleration is defined as the change in velocity per unit time. Therefore, the correct answer is acceleration.

If an object is moving with a constant velocity, it means that its speed and direction are not changing. In order for acceleration to be present, there must be a change in velocity. Since the object's velocity is not changing, the acceleration must be equal to zero.

The correct answer is a straight line making an angle with the time axis. When an object is moving with constant non-zero velocity in the +x axis, its position versus time graph will be a straight line. The angle that this straight line makes with the time axis represents the velocity of the object.

The maximum height reached by the object can be determined using the kinematic equation for vertical motion. The equation is given by:

h = (v^2 - u^2) / (2g)

Where h is the maximum height, v is the final velocity (0 m/s at the highest point), u is the initial velocity (12 m/s), and g is the acceleration due to gravity (1.5 m/s^2).

Plugging in the values, we get:

h = (0^2 - 12^2) / (2 * 1.5)
h = (-144) / 3
h = -48 m

Since height cannot be negative, the maximum height reached by the object is 48 m.

The displacement is either less than or equal to the distance traveled because displacement refers to the change in position from the starting point to the ending point, regardless of the path taken. The distance traveled, on the other hand, refers to the total length of the path taken. In some cases, the displacement may be less than the distance traveled if the object takes a longer route or changes direction multiple times. However, it can also be equal to the distance traveled if the object travels in a straight line from start to finish.

The two objects have the same initial speed, so they will experience the same acceleration due to gravity as they fall towards the ground. This means that their speeds will be the same when they hit the street. The height of the building is not relevant in determining their speeds, as it only affects the time it takes for them to reach the ground.

The acceleration of an object in free fall near the surface of the Earth is constant and equal to the acceleration due to gravity, which is approximately 9.8 m/s^2. Since both bricks are released at the same time and neglecting air resistance, they will experience the same acceleration due to gravity. Therefore, the correct answer is that the two bricks accelerate at the same rate.

The distance traveled is the sum of the distances traveled in each direction, which is 15.0 m + 11.0 m = 26.0 m. The magnitude of the displacement vector is the straight-line distance between the starting point and the ending point, which is the absolute value of the difference between the distances traveled in each direction, which is |15.0 m - 11.0 m| = 4.0 m. Therefore, the correct answer is 26.0 m, 6.0 m.

To calculate average speed, we divide the total distance traveled by the total time taken. In this case, the total distance is 350 km and the total time is 5.15 hours. Dividing 350 by 5.15 gives us approximately 67.96 km/h. Since speed is usually rounded to the nearest whole number, the correct answer is 68.0 km/h.

The team A runner can cover 100 m in 9.8 seconds, while the team B runner can cover the same distance in 10.1 seconds. To calculate the largest lead the team B runner can have, we need to find the time it takes for the team A runner to cover 100 m. Since the team A runner is the anchorman, they will start running after the other team has already started. So, the team B runner will have a head start equal to the time it takes for the team A runner to cover 100 m. The difference in time is 10.1 - 9.8 = 0.3 seconds. Using the formula speed = distance/time, we can calculate the distance covered by the team B runner in 0.3 seconds, which is 0.3 * 10 = 3.0 meters. Therefore, the largest lead the team B runner can have is 3.0 meters.

The given statement indicates that the car is slowing down, which means its velocity is decreasing. Since acceleration is the rate of change of velocity, a negative acceleration is required to cause a decrease in velocity. Therefore, the correct answer is that the car is decelerating, and its acceleration is negative.

Since both objects accelerate at the same rate, the final speed of an object is directly proportional to the time it accelerates. In this case, object A accelerates for twice the time as object B. Therefore, object A will have twice the final speed compared to object B.

The acceleration of the ball just before it reaches its highest point is exactly g. This is because at the highest point of its trajectory, the ball momentarily comes to a stop before reversing its direction and falling back down. At this point, the acceleration due to gravity is the only force acting on the ball, pulling it back towards the ground at a constant rate of g, which is approximately 9.8 m/s^2.

The average speed of the runner can be calculated by dividing the total distance covered by the time taken. In this case, the runner covered a distance of approximately 42.0 km, which is equal to 42,000 meters. The time taken was 2 hours and 57 minutes, which is equal to 177 minutes. To convert this to seconds, we multiply by 60, giving us 10,620 seconds. Dividing the distance by the time, we get an average speed of approximately 3.95 m/s.

The speed of light is given as 3.00 * 10^8 m/s. To find the distance in miles, we need to convert this speed from meters to miles. We know that 1 mile is equal to 1609 meters. Therefore, we can calculate the distance in miles by dividing the speed of light in meters by 1609.

3.00 * 10^8 m/s / 1609 m = 1.86 * 10^5 mi/s

Next, we need to find the distance traveled in one year. We know that there are 365 days in a year. Therefore, we can multiply the speed in miles per second by the number of seconds in a day (24 hours * 60 minutes * 60 seconds) and then multiply by 365.

1.86 * 10^5 mi/s * (24 * 60 * 60) s/day * 365 days/year = 5.88 * 10^12 mi

Therefore, the correct answer is 5.88 * 10^12 mi.

The average speed for the trip can be calculated by finding the total distance traveled and dividing it by the total time taken. The airplane travels 300 mi/h south for 2.00 h, covering a distance of 600 miles. Then it travels at 250 mi/h north for 750 miles. The total distance traveled is 600 + 750 = 1350 miles. The total time taken is 2.00 h + (750/250) h = 5.00 h. Therefore, the average speed for the trip is 1350 miles / 5.00 hours = 270 mi/h.

In the absence of air resistance, the only force acting on the ball is gravity. According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Since the mass of the ball remains constant and the force of gravity is constant, the acceleration of the ball remains constant as well. Therefore, the correct statement is that its acceleration is constant.

The polar bear traveled a total distance of 4.0 km (1.0 km south + 1.0 km east + 1.0 km north + 1.0 km west). Since the trip took 45 minutes, we need to convert the time to hours by dividing it by 60. Therefore, the average speed of the bear is calculated by dividing the distance (4.0 km) by the time (45/60 hours), which equals 5.3 km/h.

The average speed of a trip is calculated by dividing the total distance traveled by the total time taken. In this case, the motorist travels a total distance of 320 km (160 km + 160 km) and the total time taken is 4 hours (160 km / 80 km/h + 160 km / 100 km/h = 2 hours + 1.6 hours = 3.6 hours). Dividing the total distance by the total time gives us an average speed of approximately 89 km/h.

The car initially travels at a constant velocity of 15 m/s for 10 seconds. This means that its final velocity at the end of this time is still 15 m/s. However, after 10 seconds, the car starts to accelerate with a constant acceleration of 2.0 m/s^2 for 15 seconds. Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate the final velocity as follows: v = 15 + 2.0 * 15 = 45 m/s. Therefore, the correct answer is 45 m/s.

Given that the cart has an initial velocity of 5.0 m/s and experiences a constant acceleration of 2.0 m/s^2, we can use the equation of motion: displacement = initial velocity * time + (1/2) * acceleration * time^2. Plugging in the values, we get: displacement = 5.0 * 6.0 + (1/2) * 2.0 * (6.0)^2 = 30 + 36 = 66 m. Therefore, the cart's displacement during the first 6.0 s of its motion is 66 m.

The area under a curve in an acceleration versus time graph gives the change in velocity. This is because acceleration is the rate of change of velocity, so integrating the acceleration over time gives the change in velocity. Therefore, the correct answer is velocity.

If the velocity versus time graph of an object is a straight line making an angle of 30 degrees with the time axis, it indicates that the object is moving with a constant non-zero acceleration. This is because the slope of the velocity versus time graph represents the acceleration of the object. Since the graph is a straight line, it means that the acceleration is constant. Additionally, the fact that the line makes an angle of 30 degrees with the time axis suggests that the acceleration is non-zero.

The object is thrown upward with an initial velocity of 15 m/s. Since the acceleration due to gravity on planet X is 2.5 m/s^2, the object will experience a deceleration as it moves against the direction of gravity. The object will eventually reach its maximum height and start falling back down. The time it takes for the object to reach its maximum height can be calculated using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. In this case, the final velocity is 0 m/s (at maximum height), the initial velocity is 15 m/s, and the acceleration is -2.5 m/s^2 (opposite direction to the initial velocity). Solving for time, we get t = (v - u) / a = (0 - 15) / -2.5 = 6 s. Since the object takes the same amount of time to come back down as it took to reach the maximum height, the total time for the object to return to where it was thrown is 6 s + 6 s = 12 s.

The bear's average velocity is 0 km/h because it returns to its starting point, which means it has no displacement. Velocity is defined as the rate of change of displacement, so if there is no displacement, the velocity is 0.

The cart starts from rest and accelerates at 4.0 m/s2 for 5.0 s, which means its velocity increases by 4.0 m/s every second for 5 seconds. Therefore, the cart's velocity after the acceleration phase is 4.0 m/s * 5 s = 20 m/s.

After the acceleration phase, the cart maintains that velocity for 10 s, so its final velocity remains at 20 m/s.

Finally, the cart decelerates at a rate of 2.0 m/s2 for 4.0 s, which means its velocity decreases by 2.0 m/s every second for 4 seconds. Therefore, the cart's velocity after the deceleration phase is 20 m/s - (2.0 m/s * 4 s) = 20 m/s - 8 m/s = 12 m/s.

Thus, the final speed of the cart is 12 m/s.

The correct answer is "Yes, this is possible, and a rock thrown straight up is an example." This is because when a rock is thrown straight up, its velocity changes direction while its acceleration remains constant. Initially, the rock is accelerating upwards due to the force applied to it. As it reaches its maximum height, the velocity becomes zero, and then it starts accelerating downwards due to the force of gravity. Therefore, even though the acceleration remains constant, the velocity changes direction.

The speed of the object after 8.0 s will be 14 m/s. This is because the object is thrown upward with an initial speed of 14 m/s, and the acceleration due to gravity on planet X is 3.5 m/s^2. Since the object is thrown upward, the acceleration due to gravity will act against its motion, causing its speed to decrease. However, after 8.0 s, the object will have reached its highest point and started to fall back down. At this point, its speed will be equal to its initial speed of 14 m/s.

When an object is released from rest and falls in the absence of friction, its acceleration is constant. This is because the only force acting on the object is gravity, which causes a constant acceleration towards the ground. The velocity of the object increases continuously as it falls, but its acceleration remains constant throughout the motion.

When driving at a constant speed of 72 km/h, the distance traveled can be calculated by multiplying the speed by the time. In this case, the time is given as 4.0 seconds. To convert the speed from km/h to m/s, divide it by 3.6. So, 72 km/h is equal to 20 m/s. Multiplying the speed by the time gives us 20 m/s * 4 s = 80 m. Therefore, during the 4.0-second period of inattention, the driver travels a distance of 80 meters.

The correct answer is 9.8 m/s downward because when a ball is thrown upward, its velocity decreases due to the force of gravity. After 3.00 seconds, the ball would have reached its maximum height and started to fall back down. At this point, its velocity would be equal to the acceleration due to gravity, which is 9.8 m/s downward.

The statement "from zero to sixty in 8 s" describes the average acceleration of the car. Average acceleration is calculated by dividing the change in velocity (from zero to sixty) by the time taken (8 seconds). This description indicates how quickly the car can increase its velocity over a specific time interval, which is average acceleration.

The correct answer is that the can will have the same acceleration, both up the hill and down the hill. This is because acceleration is determined by the net force acting on an object, and in this case, the only force acting on the can is gravity. Since gravity acts in the same direction regardless of whether the can is moving up or down the hill, the acceleration remains constant.

The ratio of the maximum height of the second ball to that of the first ball is 4:1. This is because the maximum height reached by a vertically thrown ball is directly proportional to the square of its initial velocity. Since the second ball is thrown with twice the initial velocity of the first ball, its maximum height will be four times higher than that of the first ball. Therefore, the ratio is 4:1.

The object is moving with a constant acceleration, which means its velocity is changing at a constant rate. From the given information, we can calculate the acceleration by subtracting the initial velocity (16 m/s) from the final velocity (10 m/s), and dividing it by the time taken (3 seconds). This gives us an acceleration of -2 m/s^2 (negative because the velocity is decreasing).

Using the equation of motion, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can find the displacement (distance) traveled by the object during this time. Plugging in the values, we get 10 m/s = 16 m/s + (-2 m/s^2) * t. Solving for t, we get t = 3 seconds.

Now, we can use the equation of motion, s = ut + (1/2)at^2, where s is the displacement, to find the distance traveled. Plugging in the values, we get s = 16 m/s * 3 s + (1/2) * (-2 m/s^2) * (3 s)^2. Simplifying, we get s = 48 m - 9 m = 39 m. Therefore, the object moves 39 m during this time.

Based on the given information, the total time taken to drive from school to home is 4 hours and 15 minutes. Since the speed changes from 55 mi/h to 35 mi/h, it is reasonable to assume that the time taken to cover the first 110 miles is less than the time taken to cover the remaining distance. Therefore, the distance between the hometown and school can be calculated by subtracting the time taken to cover the first 110 miles from the total time, and then multiplying it by the speed of 35 mi/h. This gives us (4 hours and 15 minutes - the time taken to cover the first 110 miles) * 35 mi/h, which equals 190 miles. So, the distance between the hometown and school is 190 miles.

The correct answer is no. When the velocity of an object is zero, it does not necessarily mean that the acceleration is zero. An example that supports this is an object starting from rest. In this case, the object initially has zero velocity, but it experiences an acceleration to increase its velocity over time. Therefore, the velocity is zero, but the acceleration is not.

The length of the catapult can be determined using the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. In this case, the final velocity is 42 m/s, the initial velocity is 0 m/s (since the plane starts from rest), the time taken is 2.0 s, and the acceleration is constant. Plugging these values into the equation, we get 42 = 0 + a * 2.0. Solving for a, we find that the acceleration is 21 m/s^2. Now, using the equation s = ut + (1/2)at^2, where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken, we can find the distance traveled. Plugging in the values, we get s = 0 * 2.0 + (1/2) * 21 * (2.0)^2 = 0 + 21 * 2.0 = 42 m. Therefore, the length of the catapult is 42 m.

The object is moving with constant non-zero velocity in the +x axis, which means its velocity does not change over time. This is represented by a horizontal straight line on the velocity versus time graph. Since the velocity is constant, there is no acceleration, and therefore the graph does not show any change in velocity over time.