A robotic vehicle, or rover, is exploring the surface of Mars. The stationary Mars lander is the origin of coordinates, and the sur-rounding Martian surface lies in the xy-plane. The rover, which we x = 2.0 m - 10.25 m > s22t2represent as a point, has x- and y-coordinates that vary with time:y = 11.0 m > s2t + 10.025 m > s32t3(a) Find the rover’s coordinates and distance from the lander at t = 2.0 s. (b) Find the rover’s displacement and average velocity vectors for the interval t = 0.0 s to t = 2.0 s. (c) Find a general S. Express expression for the rover’s instantaneous velocity vector vS at t = 2.0 s in component form and in terms of magnitude and vdirection. d) Find the instantaneous acceleration at t = 2.0 s. e) a1 , t=1sec a0 , t=0sec

31) A particle has a position function r⃗ (t)=cos(1.0t)iˆ+sin(1.0t)jˆ+tkˆ,r→(t)=cos(1.0t)i^+sin(1.0t)j^+tk^, where the arguments of the cosine and sine functions are in radians. (a) What is the velocity vector? (b) What is the acceleration vector?

The velocity i of a particle moving in the xy plane is given by 7 = ( 5.50 - 5.00 ji + 9.00 3. with V in meters per second and t (> 0) in seconds. At I- 1.40 s and in unit-vector notation, what are (a) the x component and (b) the y component of the Eceleratiun? (c) Whien(if uver) is the acceluration zero? (d) At wlat pusitive time does the speed equal 10.0 mh? (a) Numher 7 Unit (b) Number 1 Unit {c) Number Unit (d) Number Unit

The position of a particle moving in the xy plane varies with time, and its coordinates are given by the following expressions: x(t) = 4.00 m +r cos[(4.00/s)t] and y(t) = r sin[(4.00/s)t], where r = 2.00 m, and x and y will be in meters when t is in seconds. Determine the following for this particle. (a) speed of the particle at any time m/s (b) magnitude of the acceleration of the particle at any time m/s?

(c)2+1)2 (t²+1) 6. Velocity of a particle depends on position as V=- dv then find dx [velocity gradient] 1-x (a) 2(1+x)² آکا x-1 (c) 1-x 2√x(1+x)² 1-x (d) 2√√x(1+x) x+1 √√x(1+x)² Intensity of electric field due to a charge distribution kr dl |= (r²+a²/3/2 where k, a are constant find dr

A particle starts from the origin of a three-dimensional coordinate system and undergoes two consecutive displacements: r 1 → = 2.0 m i ^ + 1.0 m j ^ + 3.0 m k ^  r 2 → = − 1.0 m i ^ − 3.0 m j ^ − 1.0 m k ^ What is the distance of the particle from the origin after these two displacements?

Why is (j + k) = i?I see that i x i = 0, that makes sense. But I thought that j x k = i? not j + k.

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) Explain

A particle r = (2t3+ 3t) i + (4 - 5t2) j - t4 k Position depending on time given positionmoves with the vector. Here r is in meters and t is in seconds.a) How far has it traveled at the end of the first 3 seconds?b) What is the angle between the instantaneous vector t = 2 s and the position vector?