Suppose you throw a ball into the a

Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height or to fall back to earth from its maximum height? We will solve the problem in this project but, before getting started, think about that situation and make a guess based on your physical intuition.
1. A ball with mass is projected vertically upward from the earth’s surface with a
Positive initial velocity . We assume the forces acting on the ball are the force of
gravity and a retarding force of air resistance with direction opposite to the
direction of motion and with magnitude , where is a positive constant and is the
velocity of the ball at time .
In both the ascent and the descent, the total force acting on the ball is . [During
ascent, is positive and the resistance acts downward; during descent, is negative
and the resistance acts upward.] So, by Newton’s Second Law, the equation of
motion is
Solve this differential equation to show that the velocity is
2. Show that the height of the ball, until it hits the ground, is
3. Let be the time that the ball takes to reach its maximum height. Show that
Find this time for a ball with mass 1 kg and initial velocity 20 m#s. Assume the
air resistance is of the speed.
;4. Let be the time at which the ball falls back to earth. For the particular ball in
Problem 3, estimate by using a graph of the height function . Which is faster,
going up or coming down?
5. In general, it’s not easy to find because it’s impossible to solve the equation
explicitly. We can, however, use an indirect method to determine whether ascent
or descent is faster; we determine whether is positive or negative. Show that
where . Then show that and the function is increasing for . Use this result to
decide whether is positive or negative. What can you conclude? Is ascent or
descent faster?

Kết quả (Anh) 1:

Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height or to fall back to earth from its maximum height? We will solve the problem in this project but, before getting started, think about that situation and make a guess based on your physical intuition.1. A ball with mass is projected vertically upward from the Earth's surface with a Positive initial velocity. We assume the forces acting on the ball are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude, where is a positive constant and is the velocity of the ball at time. In both the ascent and the descent, the total force acting on the ball is. [During ascent, is positive and the resistance acts downward; during descent, is negative and the resistance acts upward.] So, by Newton's Second Law, the equation of motion is Solve this differential equation to show that the velocity is2. Show that the height of the ball, until it hits the ground, is3. Let be the time that the ball takes to reach its maximum height. Show that Find this time for a ball with mass 1 kg and initial velocity 20 m # s. Assume the air resistance is of the speed.; 4. Let be the time at which the ball falls back to earth. For the particular ball in. Problem 3, the estimate by using a graph of the height function. Which is faster, going up or coming down?5. In general, it's not easy to find because it's impossible to solve the equation explicitly. We can, however, use an indirect method to determine whether ascent or descent is faster; We determine whether is positive or negative. Show that where. Then show that the function is increasing for. Use this result to decide whether is positive or negative. What can you conclude? Is ascent or descent faster?

Kết quả (Anh) 2:

Suppose you throw a ball Into the air. Do you think it takes longer to Reach its maximum height or to fall back to earth from its maximum height? We will solve the problem but in this project, all before Getting Started, think about and make a guess mà situation based on physical intuition của.
1. A ball is projected with mass vertically from the earth's surface Upward with a
Positive initial velocity. We giả sử the Forces acting on the ball are the force of
gravity and a force of air resistance Retarding with direction opposite to the
direction of motion and with magnitude, where is a positive constant and is the
velocity of the ball at time.
In cả the ascent and the descent, the total force acting on the ball is. [During
ascent, and the resistance is positive downward Acts; khi descent, is negative
and the resistance Upward Acts.] So, by Newton's Second Law, the equation of
motion is
a differential equation to show this Solve rằng velocity is
2. Show là height of the ball, an until it hits the ground, is
3. Let là là ball takes time to Reach its maximum height. Show mà
Find this time for a ball with a mass of 1 kg and initial velocity of 20 m # s. Giả sử the
air resistance is of the speed.
; 4. Let be the time at đó ball falls back to earth. For the particular ball in
Problem 3, Estimate by using a graph of the height function. Which is faster,
going up or coming down?
5. In general, it's not easy to find it's impossible to solve vì the equation
cài. We can, Tuy nhiên, use an indirect method to quyết nếu ascent
or descent is faster; chúng is positive or negative nếu quyết. Show mà
where. Then show mà and the function is tăng for. Use this result to
positive or negative is nếu decide. What can you conclude? Is ascent or
descent faster?