An ant crawls on the surface of a ball of radius b in such a manner that the ant's motion is given in spherical coordinates by the equations r = b = wt 0 = 0-1 π 1+ cos(4wt) 4 Find the speed of the ant as a function of the time t. What sort of path is represented by the above equations? [10]
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- Check Your Understanding Verify that vector v V obtained in Example 2.14 is indeed a unit vector by computing its magnitude. If the convoy in Example 2.8 was moving across a desert flatland—that is, it the third component of its velocity was zero—what is the unit vector of its direction of motion? Which geographic direction does it represent?To anticipate the dip and the hump in the road, the driver of a car applies his brakes to produce a uniform deceleration. His speed is 100kph at the bottom A of the dip and 50 kph at the top Cof the hump. The length of the road from A to Cis 120m. The radius of curvature of the hump at C is 150m. The total acceleration at A is 3 m/s^2. Compute the radius of curvature at A. C 60 m В 60 m A A) 612.15m B) 502.39m c) 432.32m D 243.23m E) 549.81mA particle describes a circumference arc of 6.6m radius. At the instant that the speed os v= 6.6m/s and the angle os 53 between the radius and the total acceleration. Calculate tangencial acceleration
- Question 6: A particle is moving in a circular path of radius R. At some in- stant, its acceleration a makes angle 0 with the horizontal as shown in Figure 2. At that instant, what is the speed of the particle? Figure 2 Select one: VaR cos 0 VaRtan 0 VaRsin 0 aR cot 0 aRsin? 0A car travels along a horizontal circular curved road that has radius of 600 m. If the speed is uniformly increased at a rate of 2000 km/h2, determine the magnitude of the acceleration (m/s2) at the instant the speed of the car is 60 km/h. Show all necessary math and include all units.three-fourths of the circular track, up to B. She then turns back and jogs one-half of 4. A jogger runs along a circular track of radius R = 100.0 m. She starts at A and jogs the circular track, up to C. (Note that the dashed line is only a guide for the eye- 4. A jogger runs along a circular track of radius R ree-fourths of the circular track, up to B. She then turns back and jogs one-half of te circular track, up to C. (Note that the dashed line is only a guide for the eye - showing that B, the center of the circle, and C are collinear). A. What is the distance she traveled? B. Draw her displacement vector. C. What is the magnitude of her displacement? C A B
- A bee flies 2 km [E] in 400s, then 4 km [E 75 degrees] in 2 000s. Calculate the speed and velocity of the trip.0, y 0) 44. A projectile is launched from the point (x QC with velocity (12.0î + 49.0 j) m/s, at t = 0. (a) Make a table listing the projectile's distance |r| from the ori- gin at the end of each second thereafter, for 0 s I S 10 s. Tabulating the x and y coordinates and the compo- nents of velocity v and v will also be useful. (b) Notice that the projectile's distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the veloc- ity. Suggestion: Argue that if v is not perpendicular to f, then r|must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) ExplainA particle moves through 3-space in such a way that its acceleration is a(t)= 8sin 2ri +8cos 2tj+e'k . where t is time measured in seconds. The initial velocity of the particle is v, = 2i – 3j+k. At time t = t , find i) the velocity vector of the particle. ii) the scalar tangential and normal components of acceleration.
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