Projectile Motion Analysis: Time in Air Calculation

[Blox_Nitrates]

Homework Statement

A projectile is fired horizontally from a gun that is 45.0m above flat ground, emerging from the gun with a speed of 250m/s. How long does the projectile remain in the air?

Homework Equations

unsure of what equation to use.

The Attempt at a Solution

I tried to use t = Δx/Vox which didn't work because it gave me 45/250=.18

*The answer is 3.03 seconds in the back of the book.

 

[CAF123]

Homework Statement

A projectile is fired horizontally from a gun that is 45.0m above flat ground, emerging from the gun with a speed of 250m/s. How long does the projectile remain in the air?

Homework Equations

unsure of what equation to use.

The Attempt at a Solution

I tried to use t = Δx/Vox which didn't work because it gave me 45/250=.18

If you want to use that eqn to find t then you will need the horizontal distance that the body travels. What other kinematic equations do you know?

 

[Blox_Nitrates]

I have these
v=vo+at
x-xo=vot+(1/2)at^2
v^2=Vo^2+2a(x-xo)
x-xo=(1/2)(vo+v)t
x-xo=vt-(1/2)at^2

Sorry about asking something probably simple as to I'm new to Calculus based Phyics and have never seen physics before.

 

[LawrenceC]

Question: If you dropped a bomb from a stationary hot air balloon while simultaneously dropping another bomb from a horizontally flying aircraft at the same altitude, would there be any difference in the amount of time it takes for the bombs to hit the ground?

 

[Blox_Nitrates]

Question: If you dropped a bomb from a stationary hot air balloon while simultaneously dropping another bomb from a horizontally flying aircraft at the same altitude, would there be any difference in the amount of time it takes for the bombs to hit the ground?

No, there wouldn't be.

 

[LawrenceC]

That is correct. Therefore you have a free fall problem. The muzzle velocity is analogous to the moving airplane. It is extraneous data that is supplied to confuse the student.

 

[Blox_Nitrates]

That is correct. Therefore you have a free fall problem. The muzzle velocity is analogous to the moving airplane. It is extraneous data that is supplied to confuse the student.

So in this case t = (V-Vo)/a = ((250m/s)/-9.8m/s^2)?

but a = 0 due to no acceleration because of horizontal motion.

 

[LawrenceC]

No, the 250 m/s is the horizontal velocity and is extraneous data. The problem is the same as if you dropped a brick off a 45 m high building. How long would it take to reach the ground?

 

[Blox_Nitrates]

t = (0-45m/s)/-9.8m/s^2 = 4.59s?

Is that correct?

 

[LawrenceC]

You are attempting to subtract meters/second from meters. You cannot do that.

What about the formula

y = Vo*t + .5*a*t^2

where Vo is initial VERTICAL velocity
a is acceleration of gravity
t is time
y is distance of free fall

 

[Blox_Nitrates]

45=0*t+.5(9.8)t^2

(45/4.9)^(1/2) = 3.03s

Thank you so much for your help LawrenceC!

So if I was asked At what horizontal distance from the firing point does it strike the ground? I would use a similar eqn to solve it while plugging in t correct?

 

[mfb]

The horizontal velocity does not change (neglecting air drag), so you can use your initial formula (in post 1) to calculate that distance.

 

[Blox_Nitrates]

The horizontal velocity does not change (neglecting air drag), so you can use your initial formula (in post 1) to calculate that distance.

I'm not sure if this is correct but I used Savg=TotalDistance/Δt
Total Distance = 250m/s(3.03s) = 758m rounded up.
The answer does match the one in the book.

 

[mfb]

That is correct.

 

[Blox_Nitrates]

This question throws me off.
A projectile's launch speed is five times its speed at maximum height. Find launch angle θo.

 

[mfb]

That is a nice question.

At maximum height, can you determine the direction of motion? Can you say something about the launch velocity components, based on that?

 

[Blox_Nitrates]

I can say that the max height will be vertical and the direction of motion is upwards. Wouldn't that be a 90 degree angle of (pi/2)?

 

[mfb]

and the direction of motion is upwards

If it would go upwards there, it would not have reached its maximal height.

Wouldn't that be a 90 degree angle of (pie/2)?

It is pi, pie is something to eat. And it depends on the definition of the angle.

 

[Blox_Nitrates]

If it would go upwards there, it would not have reached its maximal height.


It is pi, pie is something to eat. And it depends on the definition of the angle.

Sorry. So would the maximal height be somwhere between 45-90 degrees?

 

[haruspex]

Sorry. So would the maximal height be somwhere between 45-90 degrees?

It doesn't mean that the angle is chosen to achieve maximum height. The angle is whatever it is. mfb is asking what the direction of motion of the projectile will be when the projectile is at its maximum height on the trajectory. Sketching it might help.

 

[Blox_Nitrates]

It doesn't mean that the angle is chosen to achieve maximum height. The angle is whatever it is. mfb is asking what the direction of motion of the projectile will be when the projectile is at its maximum height on the trajectory. Sketching it might help.

I would see a parabolic function in that case but I still feel like I'm not seeing something that I'm possibly missing.

 

[haruspex]

I would see a parabolic function in that case but I still feel like I'm not seeing something that I'm possibly missing.

That's right. So which way is the projectile moving when at max height?

 

[Blox_Nitrates]

That's right. So which way is the projectile moving when at max height?

Somewhat angled about 60 degrees moving upward, hitting the apex when a=0 and then moving downward towards the ground.

 

[haruspex]

Somewhat angled about 60 degrees moving upward,

If it's moving upward it's not yet at max height.

hitting the apex when a=0

That's max height of the trajectory. Which way is it moving there?

 

[Blox_Nitrates]

If it's moving upward it's not yet at max height.

That's max height of the trajectory. Which way is it moving there?

It would be moving horizontally for an instant before moving downwards.

 

[haruspex]

It would be moving horizontally

Yes. Now, suppose it has speed v there. What then can you say then about its components of velocity at launch?

 

[Blox_Nitrates]

V=Vo+at?

 

[haruspex]

V=Vo+at?

What horizontal acceleration is there (if air resistance is ignored)?

 

[Blox_Nitrates]

What horizontal acceleration is there (if air resistance is ignored)?

There wouldn't be any acceleration right? But then that would leave V=Vo which is true?

 

[haruspex]

There wouldn't be any acceleration right? But then that would leave V=Vo which is true?

That depends what you mean by Vo. Is that the magnitude of the initial velocity, or of just a component of it?

 

[Blox_Nitrates]

The magnitude of the initial velocity.

 

[haruspex]

The magnitude of the initial velocity.

No, that's not how it works. Consider horizontal and vertical components of velocity separately. The vertical component is subject to acceleration by gravity; the horizontal one is not. So, ignoring air resistance, the horizontal component of velocity does not change. If it is V at max height, what is it at launch?

 

[Blox_Nitrates]

The horizontal component would also be V at the initial launch if ignoring air resistance because velocity wouldn't change.

 

[Blox_Nitrates]

Since we are ignoring air resistance than the initial speed v would be the same as initial speed v at max height because the horizontal component's velocity doesn't change. The speed at launch would also be speed V.

 

[haruspex]

Since we are ignoring air resistance than the initial speed v would be the same as initial speed v at max height because the horizontal component's velocity doesn't change. The speed at launch would also be speed V.

The horizontal component of the velocity does not change, but the projectile's speed (which combines horizontal and vertical components) does.
Suppose the launch speed is u and launch angle is θ above horizontal. What is the horizontal component at launch?