Chapter 3: Linear Motion

The car's acceleration can be calculated using the formula: acceleration = (final velocity - initial velocity) / time. In this case, the final velocity is 20 m/s, the initial velocity is 0 m/s (since the car starts from rest), and the time is 5 seconds. Plugging these values into the formula, we get: acceleration = (20 m/s - 0 m/s) / 5 s = 4 m/s^2. Therefore, the car's acceleration is 4 meters per second per second.

The acceleration of a car can be calculated by dividing the change in velocity by the time taken. In this case, the car increases its velocity from zero to 60 km/h in 10 seconds. The change in velocity is 60 km/h - 0 km/h = 60 km/h. Dividing this by the time taken, 10 seconds, we get an acceleration of 6 km/h/s.

The car's acceleration can be calculated using the formula: acceleration = (final velocity - initial velocity) / time. In this case, the final velocity is 40 m/s, the initial velocity is 0 m/s (since the car started from rest), and the time is 10 seconds. Plugging these values into the formula, we get acceleration = (40 m/s - 0 m/s) / 10 s = 4 m/s^2. Therefore, the car's acceleration is 4.0 meters per second per second.

The car is accelerating at a rate of 2 meters per second per second. This means that its speed is increasing by 2 meters per second every second. If the car starts from rest, it needs to reach a speed of 30 m/s. Since the rate of acceleration is constant, we can use the equation v = u + at, where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time. Plugging in the values, we get 30 = 0 + 2t. Solving for t, we find that t = 15 seconds. Therefore, it takes 15 seconds for the car to accelerate to a speed of 30 m/s.

The average speed is calculated by dividing the total distance traveled by the time taken to travel that distance. Therefore, the two measurements necessary for calculating average speed are distance and time.

The car is accelerating at a constant rate of 2 meters per second per second. This means that its speed is increasing by 2 meters per second every second. After 3 seconds, the car would have increased its speed by 2 meters per second three times, resulting in a speed of 6 meters per second.

According to the laws of physics, an object falling freely under the influence of gravity will continuously accelerate. The acceleration due to gravity is approximately 9.8 m/s². Therefore, after twelve seconds, the object would have been accelerating for a significant amount of time, resulting in a speed greater than 100 m/s.

The acceleration of the bullet is 9.8 meters per second per second because when an object is in free fall near the surface of the Earth, it experiences a constant acceleration due to gravity of approximately 9.8 m/s^2. This means that the bullet's velocity will increase by 9.8 meters per second every second it falls.

On a planet where the acceleration due to gravity is 20 m/s/s, the speed of a freely falling object would increase by 20 m/s every second. This is because the acceleration due to gravity determines how quickly an object's speed changes over time. In this case, the acceleration is constant at 20 m/s/s, so the speed reading would increase by 20 m/s each second. The initial speed of the object does not affect the rate at which its speed increases, so the correct answer is 20 m/s.

The object falls freely under the influence of gravity, so its velocity will increase at a constant rate. The acceleration due to gravity on this planet is 20 m/s^2, which means that every second, the object's velocity will increase by 20 m/s. After 5 seconds, the object's velocity will be 20 m/s * 5 s = 100 m/s.

Each bullet experiences the same acceleration of 9.8 meters per second per second due to gravity, regardless of whether it is dropped or fired downward. This is because the acceleration due to gravity is a constant value on Earth. The initial velocity of the bullets does not affect their acceleration. Therefore, both bullets experience the same acceleration of 9.8 meters per second per second.

Objects fall with constant acceleration. This means that the speed at which an object falls increases at a constant rate with each passing second. This is due to the force of gravity acting on the object, causing it to accelerate towards the ground. The acceleration due to gravity on Earth is approximately 9.8 meters per second squared. Therefore, objects do not fall with constant velocity or speed, as these would imply no change in motion. Additionally, objects do not fall with constant distances each successive second, as their displacement increases with time due to the acceleration.

The acceleration of the rocket is given as 50 m/s^2. This means that every second, the rocket's speed increases by 50 m/s. Since the rocket accelerates for one minute, which is equal to 60 seconds, the total increase in speed would be 50 m/s * 60 s = 3000 m/s. Therefore, the speed of the rocket after one minute would be 3000 m/s.

The average speed of an object is calculated by dividing the total distance traveled by the time taken. In this case, the horse galloped a distance of 10 kilometers in a time of 30 minutes. To convert the time to hours, we divide by 60 (30 minutes = 0.5 hours). Therefore, the average speed is 10 km / 0.5 hours = 20 km/h.

When an object near the Earth's surface is in free fall, it experiences a constant acceleration due to gravity. As time goes on, the object's velocity increases because the acceleration causes it to speed up. This is because the force of gravity is constantly pulling the object downward, causing it to gain speed as it falls. Therefore, the correct answer is that the velocity increases.

If the acceleration on the distant planet is twice that of Earth, it means that the object will experience a greater force pulling it down. According to the equations of motion, the speed of a freely falling object increases at a constant rate. In one second, the object's speed would be equal to the acceleration multiplied by the time, which is 2 * 1 = 2 m/s. Therefore, the object's speed one second later would be 2 m/s, which corresponds to the answer 20 m/s.

If ice friction and air resistance are neglected, there are no external forces acting on the hockey puck once it is set in motion. According to Newton's first law of motion, an object will continue to move at a constant velocity unless acted upon by an external force. Therefore, no force is required to keep the puck sliding at a constant velocity.

The distance traveled by an object falling freely would be greater than the second before because the object is accelerating due to the force of gravity. As time progresses, the object's velocity increases, causing it to cover a greater distance in each subsequent second. This is in accordance with the laws of motion and gravity.

The correct answer is 10 m/s. When a freely falling object is equipped with a speedometer, its speed reading would increase by about 10 m/s each second. This is because objects in free fall experience a constant acceleration due to gravity, which is approximately 9.8 m/s^2. Therefore, the speed of the object would increase by 9.8 m/s every second. Rounded to the nearest whole number, this would be approximately 10 m/s.

Based on the given information, we know that it takes 6 seconds for the stone to fall to the bottom of the mine shaft. Using the equation of motion, s = ut + (1/2)at^2, where s is the distance, u is the initial velocity (which is 0 in this case), t is the time, and a is the acceleration due to gravity, we can calculate the distance. Since the stone is falling, the acceleration due to gravity is -9.8 m/s^2. Plugging in the values, we get s = 0 + (1/2)(-9.8)(6^2) = 0 + (-4.9)(36) = -176.4 m. However, distance cannot be negative, so we take the magnitude of -176.4 m, which is about 176.4 m. Therefore, the depth of the shaft is about 180 m.

When an object is dropped, it accelerates due to the force of gravity. The acceleration is constant and equal to 9.8 m/s² near the surface of the Earth. In this case, the rock is dropped from a height of 5 m and accelerates at 10 m/s², which is twice the acceleration due to gravity. When the same rock is dropped from a height of 2.5 m, it will still experience the same acceleration due to gravity, which is 9.8 m/s². Therefore, the acceleration of fall for the rock is the same amount in both cases.

The correct answer is "increasing". This means that the distance a freely falling object will fall in each second of fall is increasing. This is because the object accelerates due to the force of gravity. As time passes, the object's speed and therefore the distance it falls in each second increases.

The acceleration just before striking the water is the same for each bullet. This is because both bullets are subject to the same gravitational acceleration, regardless of how they were initially propelled. Neglecting air resistance, all objects near the surface of the Earth experience the same acceleration due to gravity, which is approximately 9.8 m/s². Therefore, both the dropped bullet and the fired bullet will have the same acceleration just before striking the water.

The distance a freely falling object will fall in 0.5 seconds is not 1 meter or 10 meters. The correct answer is "none of the above" because the distance an object falls in a given time depends on the acceleration due to gravity, which is approximately 9.8 meters per second squared. Using the equation d = 0.5 * g * t^2, where d is the distance, g is the acceleration due to gravity, and t is the time, we can calculate that the distance fallen in 0.5 seconds is approximately 1.225 meters. Therefore, none of the given options accurately represent the distance fallen in 0.5 seconds.

The apple falls from a tree, which means it is subject to the force of gravity pulling it downwards. As it falls, it accelerates due to this force. The speed at which it hits the ground can be estimated using the equation v = sqrt(2gh), where v is the final velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height from which the apple fell (5 meters in this case). Plugging in the values, we get v = sqrt(2 * 9.8 * 5) ≈ 10 m/s. Therefore, the correct answer is 10 m/s.

The time it takes for an object to fall is determined by the height from which it falls and the acceleration due to gravity. In this case, since it takes 6 seconds for the stone to fall, we can use 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. Plugging in the values, we get h = (1/2)(9.8)(6)^2 = 176.4 m. Therefore, the depth of the shaft is approximately 180 m.

Since the car maintains a constant velocity of 100 km/hr for 10 seconds, it means that there is no change in its velocity over time. Acceleration is defined as the rate of change of velocity, so if there is no change in velocity, the acceleration must be zero.

When the ball is thrown upwards and returns to the same position, its speed when it returns is the same as its original speed after release. This is because the gravitational force acting on the ball causes it to decelerate as it moves upwards, eventually bringing it to a stop. As the ball starts to fall back down, it accelerates due to the gravitational force, and its speed increases. However, at the same height from which it was released, its speed is the same as its original speed after release. Therefore, the correct answer is "the same."

The object covers the same distance of 8 meters in each second of travel, indicating that its velocity remains constant. Since acceleration is the rate of change of velocity, and the velocity does not change, the acceleration is 0.

When the man throws the rock upward, it will initially move against gravity. After 2 seconds, the rock will reach its highest point and start to fall back down due to gravity. The distance the rock falls during this time is equal to the distance it was thrown upward. Since the initial velocity of the rock was 4.9 m/s and it fell for 2 seconds, the distance it fell is given by the equation d = 1/2 * g * t^2, where g is the acceleration due to gravity (9.8 m/s^2) and t is the time (2 seconds). Plugging in the values, we get d = 1/2 * 9.8 m/s^2 * (2 s)^2 = 9.8 m. Therefore, the rock is 9.8 m below the level from which it was thrown.

When an object is dropped, it falls freely under the influence of gravity. The acceleration due to gravity is constant and equal to 9.8 meters per second per second. When an object is thrown downward, it still experiences the same acceleration due to gravity. Therefore, the object's acceleration, in the absence of air resistance, will also be 9.8 meters per second per second.

When an apple falls from a tree, it accelerates due to the force of gravity. The acceleration due to gravity is approximately 9.8 m/s². Using the equation v² = u² + 2as, where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and s is the distance fallen, we can calculate the final velocity. Plugging in the values, we get v² = 0 + 2 * 9.8 * 5, which simplifies to v² = 98. Solving for v, we find that the final velocity is approximately 9.9 m/s. Since the question asks for an estimate, the closest option is 10 m/s.

When an object is freely falling, it accelerates due to gravity at a rate of 9.8 m/s². After half a second, the object would have been accelerating for 0.5 seconds. Using the equation v = u + at, where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time, we can calculate the final velocity. Plugging in the values, we get v = 0 + (9.8 * 0.5) = 4.9 m/s. Therefore, the speed of the object after half a second would be approximately 5 m/s.

To calculate the speed at which the ball must be tossed, we can use the equation of motion for vertical motion. The time taken for the ball to reach its highest point and return to the same level is the total time of 2 seconds. Since the ball is thrown straight up, the initial velocity is positive and the final velocity when it returns is negative. Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration (which is -9.8 m/s^2 for objects moving upwards), and t is the time, we can substitute the given values. Rearranging the equation, we get u = v - at. Substituting v = -u, t = 2 seconds, and a = -9.8 m/s^2, we can solve for u. u = -(-u) - (-9.8 m/s^2)(2 s). Simplifying the equation, we get 2u = 19.6 m/s^2. Solving for u, we find u = 9.8 m/s. Therefore, the ball must be tossed with a speed of 10 m/s upwards to take 2 seconds to return to the starting level.

The pot falls from a ledge and hits the ground 45 m below. The speed with which it hits the ground can be calculated using the equation for free fall. The equation is given by v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration due to gravity (approximately 9.8 m/s^2), and s is the distance fallen (45 m). Plugging in the values, we get v^2 = 0 + 2(9.8)(45), which simplifies to v^2 = 882. Taking the square root of both sides, we get v ≈ 29.7 m/s. Therefore, the speed with which the pot hits the ground is about 30 m/s.

The correct answer is "increasing" because as an object falls freely, its distance of fall increases with each passing second. This is due to the acceleration of gravity, which causes the object to gain speed and cover more distance in each subsequent second. Therefore, the distance a freely falling object will fall is not a constant value like 5 m or 10 m, but rather it increases over time.

When an object is in free fall, its speed increases by approximately 9.8 meters per second every second due to the acceleration due to gravity. Therefore, one second later, the object's speed should increase by 9.8 m/s, resulting in a speed of approximately 59.8 m/s. Since this is closest to 60 m/s, the correct answer is 60 m/s.

If an object moves with constant acceleration, its velocity must change by the same amount each second. This is because acceleration is defined as the rate of change of velocity. If the acceleration is constant, it means that the object's velocity is changing at a constant rate. Therefore, the change in velocity will be the same for each unit of time, in this case, each second.

The car accelerates at a constant rate of 2 meters per second squared. This means that its velocity increases by 2 meters per second every second. Since the car starts from rest, its initial velocity is 0 meters per second. Using the equation d = v0t + (1/2)at^2, where d is the distance traveled, v0 is the initial velocity, t is the time, and a is the acceleration, we can calculate the distance traveled. Plugging in the values, we get d = 0(10) + (1/2)(2)(10)^2 = 0 + 10(10) = 100 meters. Therefore, the car will travel 100 meters in 10 seconds.

The object is moving upward at a speed of 50 m/s. However, one second later, the object would experience the force of gravity acting on it, causing its velocity to decrease. Therefore, its speed would be less than 50 m/s. The closest option is 40 m/s, which would be the approximate speed of the object one second later.

The time it takes for a projectile to return to its starting position can be estimated by considering the motion of the object. When a projectile is fired straight up, it will reach its highest point (the peak of its trajectory) and then fall back down. The time it takes for the projectile to reach its highest point is half of the total time it takes to return to its starting position. Therefore, if the total time is estimated to be 2 seconds, it means that the projectile takes 1 second to reach its highest point and another 1 second to fall back down, resulting in a total time of 2 seconds.

When a rock thrown straight upwards gets to the exact top of its path, its velocity is zero because it momentarily stops moving before it starts falling back down. Its acceleration is about 10 meters per second per second because it experiences a constant acceleration due to gravity pulling it downwards. This means that its speed decreases at a rate of 10 meters per second every second until it reaches the top of its path.

Both balls will hit the ground with the same speed because the initial speed at which they were thrown is the same. The only difference is the direction in which they were thrown, but gravity affects both balls equally and accelerates them at the same rate. Therefore, they will both hit the ground with the same speed.

When a projectile is fired straight up, its initial velocity is in the opposite direction of gravity. As the projectile moves upward, gravity acts as a decelerating force, eventually bringing it to a stop at the top of its path. At this point, the projectile starts to fall back down. The time it takes to reach the top of its path can be estimated by considering the initial velocity and the acceleration due to gravity. Since the initial velocity is 10 m/s and the acceleration due to gravity is approximately 9.8 m/s^2, it takes approximately 1 second for the projectile to reach the top of its path.

As the drops of water fall from a dripping faucet, they experience air resistance and gravity. Over time, the air resistance causes the drops to slow down, and gravity pulls them downward. This combination of forces causes the drops to gradually increase the distance between each other as they fall, resulting in them getting farther apart.

When a ball is thrown upward and then falls back to Earth, it follows a parabolic trajectory. The time it takes for the ball to reach its maximum height is equal to the time it takes for it to fall back down to the ground. This means that the time spent going up is equal to the time spent going down. Since the total distance traveled by the ball is 125 meters, and it spends an equal amount of time going up and coming down, the total time in the air would be approximately 10 seconds.

The correct answer is (100 - 4.9) m. This is because the bullet is fired straight up into the air, meaning it is subject to the force of gravity pulling it downwards. The bullet will experience a constant acceleration of 9.8 m/s^2 due to gravity. Using the equation of motion, s = ut + (1/2)at^2, where s is the distance traveled, u is the initial velocity, t is the time, and a is the acceleration, we can calculate the distance traveled at the end of one second. Plugging in the values, s = 100(1) + (1/2)(-9.8)(1)^2 = 100 - 4.9 = 95.1 m.

To remain in the air for a total time of 10 seconds, the ball needs to reach its maximum height and then fall back down within that time frame. The time it takes for the ball to reach its maximum height is half of the total time, so 10/2 = 5 seconds. Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration (which is equal to -9.8 m/s^2 due to gravity), and t is the time, we can solve for the initial upward speed. Plugging in the values, we get 0 = u - 9.8 * 5. Solving for u, we find that the initial upward speed should be approximately 49 m/s, which is closest to 50 m/s.

When the man throws the rock upward at 4.9 m/s, it starts moving against the force of gravity. As time passes, the rock's upward velocity decreases due to the gravitational pull. After one second, the rock's speed will decrease to zero and then start falling back down. Therefore, the correct answer is 4.9 m/s, as it represents the initial velocity with which the rock was thrown.

The acceleration of the ball is directed downward because it is always acting in the opposite direction of the ball's velocity. When the ball is tossed upward, its velocity is initially directed upward, but as it rises, its velocity decreases until it reaches its highest point. At this point, the velocity is momentarily zero, but the acceleration is still directed downward due to the force of gravity. As the ball falls back to its starting point, its velocity becomes negative (directed downward) and the acceleration remains directed downward as well.