The component that allows a car to execute complex maneuvers is __________ .

1 answer:

6 0

The anti-lock braking system (ABS) is a car component that enables the driver to execute complex maneuvers. Furthermore, it allows shifting the position of your car when driving on highways because of its ability to "maintain traction." It also can decrease the kinetic energy of your car when you want to make it decelerated.

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A projectile of mass m is launched with an initial velocity vector v i making an angle θ with the horizontal as shown below. The

sergeinik [125]

Angular momentum is given by the length of the arm to the object, multiplied by the momentum of the object, times the cosine of the angle that the momentum vector makes with the arm. From your illustration, that will be:
L = R * m * vi * cos(90 - theta) 

cos(90 - theta) is just sin(theta) 
and R is the distance the projectile traveled, which is vi^2 * sin(2*theta) / g 

so, we have: L = vi^2 * sin(2*theta) * m * vi * sin(theta) / g 

We can combine the two vi terms and get: 

L = vi^3 * m * sin(theta) * sin(2*theta) / g 

What's interesting is that angular momentum varies with the *cube* of the initial velocity. This is because, not only does increased velocity increase the translational momentum of the projectile, but it increase the *moment arm*, too. Also note that there might be a trig identity which lets you combine the two sin() terms, but nothing jumps out at me right at the moment. 

Now, for the first part... 

There are a few ways to attack this. Basically, you have to find the angle from the origin to the apogee (highest point) in the arc. Once we have that, we'll know what angle the momentum vector makes with the moment-arm because, at the apogee, we know that all of the motion is *horizontal*. 

Okay, so let's get back to what we know: 

L = d * m * v * cos(phi) 

where d is the distance (length to the arm), m is mass, v is velocity, and phi is the angle the velocity vector makes with the arm. Let's take these one by one... 

m is still m. 
v is going to be the *hoizontal* component of the initial velocity (all the vertical component got eliminated by the acceleration of gravity). So, v = vi * cos(theta) 
d is going to be half of our distance R in part two (because, ignoring friction, the path of the projectile is a perfect parabola). So, d = vi^2 * sin(2*theta) / 2g 

That leaves us with phi, the angle the horizontal velocity vector makes with the moment arm. To find *that*, we need to know what the angle from the origin to the apogee is. We can find *that* by taking the arc-tangent of the slope, if we know that. Well, we know the "run" part of the slope (it's our "d" term), but not the rise. 

The easy way to get the rise is by using conservation of energy. At the apogee, all of the *vertical* kinetic energy at the time of launch (1/2 * m * (vi * sin(theta))^2 ) has been turned into gravitational potential energy ( m * g * h ). Setting these equal, diving out the "m" and dividing "g" to the other side, we get: 

h = 1/2 * (vi * sin(theta))^2 / g 

So, there's the rise. So, our *slope* is rise/run, so 

slope = [ 1/2 * (vi * sin(theta))^2 / g ] / [ vi^2 * sin(2*theta) / g ] 

The "g"s cancel. Astoundingly the "vi"s cancel, too. So, we get: 

slope = [ 1/2 * sin(theta)^2 ] / [ sin(2*theta) ] 

(It's not too alarming that slope-at-apogee doesn't depend upon vi, since that only determines the "magnitude" of the arc, but not it's shape. Whether the overall flight of this thing is an inch or a mile, the arc "looks" the same). 

Okay, so... using our double-angle trig identities, we know that sin(2*theta) = 2*sin(theta)*cos(theta), so... 

slope = [ 1/2 * sin(theta)^2 ] / [ 2*sin(theta)*cos(theta) ] = tan(theta)/4 

Okay, so the *angle* (which I'll call "alpha") that this slope makes with the x-axis is just: arctan(slope), so... 

alpha = arctan( tan(theta) / 4 ) 

Alright... last bit. We need "phi", the angle the (now-horizontal) momentum vector makes with that slope. Draw it on paper and you'll see that phi = 180 - alpha 

so, phi = 180 - arctan( tan(theta) / 4 ) 

Now, we go back to our original formula and plug it ALL in... 

L = d * m * v * cos(phi) 

becomes... 

L = [ vi^2 * sin(2*theta) / 2g ] * m * [ vi * cos(theta) ] * [ cos( 180 - arctan( tan(theta) / 4 ) ) ] 

Now, cos(180 - something) = cos(something), so we can simplify a little bit... 

L = [ vi^2 * sin(2*theta) / 2g ] * m * [ vi * cos(theta) ] * [ cos( arctan( tan(theta) / 4 ) ) ]

3 0

3 years ago

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A man, a distance d=3~\text{m}d=3 m from a target, throws a ball at an angle \theta= 70^\circθ=70 ∘ above the horizontal. If

lbvjy [14]

Answer:

The ball doesn't strike the building because it strikes the ground at d=1.62 meters.

Explanation:

V= 5 m/s < 70º

Vx= 1.71 m/s

Vy= 4.69 m/s

h= Vy * t - g * t²/2

clearing t for the flying time of the ball:

t= 0.95 s

d= Vx * t

d= 1.62 m

4 0

3 years ago

Electrons in an atom make up most of the _________ of the atom, while the protons and neutrons make up nearly all of the _______

Vinvika [58]

Answer:

volume;mass

Explanation:

3 0

3 years ago

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A ball traveling at a speed ν0 rolls off a desk and lands at a horizontal distance x0 away from the desk, as shown in the figure

klasskru [66]

Answer:

$3x_0$

Explanation:

The horizontal distance covered by the ball in the falling is only determined by its horizontal motion - in fact, it is given by

$d=v_x t$

where

$v_x$ is the horizontal velocity

t is the time of flight

The time of flight, instead, is only determined by the vertical motion of the ball: however, in this problem the vertical velocity is not changed (it is zero in both cases), so the time of flight remains the same.

In the first situation, the horizontal distance covered is

$d=v_0 t = x_0$