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Calc 3 10.9

 

<== Calculus 3
Number Question Answer
See the work
1. Find the velocity, acceleration, and speed of a particle with position function
\mathbf r(t) = \langle -8 t \sin t, -8 t \cos t, 8 t^2
angle
v(t) = &lang,, &rang
a(t) = &lang,, &rang
| v(t) | =
v(t) =
&lang , , 16t &rang
a(t) =
&lang,,16&rang
| v(t) | =
The work
2. Find the velocity and position vectors of a particle with acceleration a(t) = &lang 0,0,4 &rang, and initial conditions v(0) = &lang 3, -3, -5 &rang and r(0) = &lang 3, -2, -3 &rang .
v(t) = &lang, , &rang
r(t) = &lang, ,&rang
v(t) = &lang 3, -3, 4t-5 &rang
r(t) = &lang 3t+3, -3t-2, &rang
The work
3. Given that the acceleration vector is a(t) = ( -cos(t) )i - ( sin(t) )j + (2t)k, the initial velocity is v(0) = i + k, and the initial position vector is r(0) = i + j + k, compute:

A. The velocity vector v(t) = i + j + k

B. The position vector r(t) = i + j + k

v(t) = (-sin(t) + 1)i + (cos(t ) - 1)j + k

r(t) = (cos( t ) + t)i + (sin( t ) -t + 1)j + ()k

The work
4. The position function of a particle is given by \mathbf r(t) = \langle 1 t^2, 5 t, t^2  + 2 t
angle.
At what time is the speed minimum?
-0.5 The work
5. A dense particle with mass 7 kg follows the path \mathbf{r}(t) = (\sin(8 t), \cos(8 t) , 2t^{7/2}) with units in meters and seconds.
What force acts on the mass at t = 0?

(, , ) kg*m/s^2

( 0, -448, 0) kg*m/s^2 The work
6. A projectile is fired from ground level with an initial speed of 550 m/sec and an angle of elevation of 30 degrees. Use that the acceleration due to gravity is 9.8 m/sec^2.
The range of the projectile is meters.
The maximum height of the projectile is meters.
The speed with which the projectile hits the ground is m/sec.
Range of projectile = 26731.91 meters.
Max height of projectile = 3858.42 meters.
Speed projectile hits ground = 550 m/sec.
The work
7. A ball is thrown at an angle of 45 degrees to the ground, and lands 40 meters away.
The initial speed of the ball was m/sec.
19.799 m/sec The work
8. A body of mass 10 kg moves in a (counterclockwise) circular path of radius 8 meters, making one revolution every 12 seconds. You may assume the circle is in the xy-plane, and so you may ignore the third component.
A. Compute the centripetal force acting on the body.
(,)
B. Compute the magnitude of that force.
A) (-21.932cos(0.5236t), -21.932sin(0.5235t))

B) 21.932
The work