All categories

3 years ago

We consider a projectile motion against a linear drag force D = −b∗v, where v is the velocity

Physics

1 answer:

Alex3 years ago

6 0

Explanation:

(A) Sum of forces on the projectile in the y direction:

-bv − mg = ma

Acceleration is the derivative of velocity with respect to time:

-bv − mg = m dv/dt

Separate the variables:

bv + mg = -m dv/dt

-1/m dt = 1/(bv + mg) dv

-b/m dt = b/(bv + mg) dv

Integrate:

-b/m t |₀ᵗ = ln(bv + mg) |₀ᵛ

-b/m (t − 0) = ln(bv + mg) − ln(0 + mg)

-b/m t = ln(bv + mg) − ln(mg)

-b/m t = ln((bv + mg) / mg)

e^(-b/m t) = (bv + mg) / mg

bv + mg = mg e^(-b/m t)

bv = -mg + mg e^(-b/m t)

v = -mg/b (1 − e^(-b/m t))

Velocity is derivative of position with respect to time:

dz/dt = -mg/b (1 − e^(-b/m t))

Separate the variables:

-b/(mg) dz = (1 − e^(-b/m t)) dt

Integrate:

-b/(mg) z |ᵧᶻ = (t + m/b e^(-b/m t)) |₀ᵗ

-b/(mg) (z − h) = (t + m/b e^(-b/m t)) − (0 + m/b e^(0))

-b/(mg) (z − h) = t + m/b e^(-b/m t) − m/b

z − h = -mg/b (t + m/b e^(-b/m t) − m/b)

z = h − mg/b (t + m/b e^(-b/m t) − m/b)

(B) Repeat steps from part A, but this time in the x direction.

-bv = ma

-bv = m dv/dt

-b/m dt = 1/v dv

-b/m t |₀ᵗ = ln v |ᵥᵛ

-b/m (t − 0) = ln vₓ − ln v₀ₓ

-b/m t = ln (vₓ / v₀ₓ)

vₓ / v₀ₓ = e^(-b/m t)

vₓ = v₀ₓ e^(-b/m t)

dx/dt = v₀ₓ e^(-b/m t)

dx = v₀ₓ e^(-b/m t) dt

x |₀ˣ = -m/b v₀ₓ e^(-b/m t) |₀ᵗ

x − 0 = -m/b v₀ₓ e^(-b/m t) − (-m/b v₀ₓ e^(0))

x = -m/b v₀ₓ e^(-b/m t) + m/b v₀ₓ

x = m/b v₀ₓ (1 − e^(-b/m t))

To find z(x), find t in terms of x then substitute into z(t).

b x / (m v₀ₓ) = 1 − e^(-b/m t)

e^(-b/m t) = 1 − b x / (m v₀ₓ)

-b/m t = ln(1 − b x / (m v₀ₓ))

t = -m/b ln(1 − b x / (m v₀ₓ))

z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) + m/b (1 − b x / (m v₀ₓ)) − m/b)

z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) + m/b − x / v₀ₓ − m/b)

z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)

The range is when z = 0:

0 = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)

h = mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)

bh/(mg) = -m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ

-(b/m)² h/g = ln(1 − (b/m) x / v₀ₓ) + (b/m) x / v₀ₓ

Unfortunately, this can't be simplified further without using something called the Lambert W function.

x = v₀ₓ √(2h/g)

You might be interested in

Bill was really bored at work. During his break, he decided to check his carotid artery. He placed both fingers on his neck and

Sergeeva-Olga [200]

Answer:

72 beats per minute

Explanation:

Heart beat causes the flow of blood round the body. This heart beat can be felt as pulse in the wrist or neck carotid artery. The heart rate which is measured in beats per minute (BPM) is used to determine the number of heart beats per minute.

You can calculate your BPM using the carotid artery found in the neck close to the windpipe.

Given that for 20 seconds, Bill had a total of 24 beats.

60 seconds = 1 minute.

Hence, Bill's BPM = (24 beats per 20 seconds) * (60 seconds per minute) = 72 beats per minute

5 0

3 years ago

A brick of mass 2.0kg is at rest. It falls to the ground through a

Lisa [10]

Answer:

I may not have the answer so i'll just give up some hints.

Multiply the time by the acceleration due to gravity to find the velocity when the object hits the ground. If it takes 9.9 seconds for the object to hit the ground, its velocity is (1.01 s)*(9.8 m/s^2), or 9.9 m/s. Choose how long the object is falling. In this example, we will use the time of 8 seconds. Calculate the final free fall speed (just before hitting the ground) with the formula v = v₀ + gt = 0 + 9.80665 * 8 = 78.45 m/s . Find the free fall distance using the equation s = (1/2)gt² = 0.5 * 9.80665 * 8² = 313.8 m .h = 0.5 * 9.8 * (1.5)^2 = 11m. b. V = gt = 9.8 * 1.5 = 14.7m/s. A feather and brick dropped together. Air resistance causes the feather to fall more slowly. If a feather and a brick were dropped together in a vacuum—that is, an area from which all air has been removed—they would fall at the same rate, and hit the ground at the same time.When an object's point is taller the thing that is going down it will go faster than when the point is lower. EXAMPLE: The object is the tennis ball if you drop it down the higher hill it will be faster than if you drop it down a shorter hill. In other words, if two objects are the same size but one is heavier, the heavier one has greater density than the lighter object. Therefore, when both objects are dropped from the same height and at the same time, the heavier object should hit the ground before the lighter one.

I hope my little bit (big you may say) hint help you with your question.

5 0

3 years ago

I am sitting on a train car traveling horizontally at a constant speed of 50 m/s. I throw a ball straight up into the air.

drek231 [11]

Answer:

Explanation:

I am sitting on a train car traveling horizontally at a constant speed of 50 m/s. I throw a ball straight up into the air. Before , the ball gets separated from my hand , both me the ball will be moving with velocity of 50 m /s in horizontal direction .

As soon as ball is separated from the hand , it acquires addition velocity in upward direction and acceleration in downward direction . This will give relative velocity to the ball with respect to me . So I will see the ball going in upward direction under gravitational acceleration . It appears as if I am sitting at rest and ball is going in upward direction under deceleration . My motion at 50 m/s will have no effect on the motion of ball in upward direction , according to first law of Newton . It is so because ball too will be moving in forward direction with the same speed which will not be visible to me because I too am moving with the same speed.

If I am sitting at rest at home and I threw a ball straight up into the air , I will have the same experience of seeing ball going in similar way as described above.

8 0

3 years ago

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then rel

Svetllana [295]

<span>To get that, find the time when slope equals to 0, use the derivate theory from calculus.

s = 8 cos t + 9 sin t, t ≥ 0
s' = -8 sin t + 9 cos t (this is slope)
0 = -8 sin t + 9 cos t
8 sin t = 9 cos t
tan t = 9/8
t = 48.37 seconds</span>

4 0

3 years ago

What happens when the temperature of a substance is increased?

Amiraneli [1.4K]

When the temperature of a substance is increased, the kinetic energy of the particles of the substance is increased as well.

This can lead to thermal expansion such as an increase in length of the substance or increase in volume of the substance.

This can also lead to change of state of the substance, for example change of state from solid to liquid or from liquid to vapor.
<span />

7 0

4 years ago