Recall Newton's Second Law relating the acceleration of an object to its mass: F a m Suppose an object that masses 7 kilograms is at the origin (0,0,0) and given an initial velocity of Vo = (-1,0,1). Then a force with magnitude 40 Newtons acts on this object in the positive z-direction. Find this object's position function r(t). You should assume the force begins acting at time t-0. Submit a clear, detailed, and complete solution.
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- A particle is moving along a line with position given by s(t)=t3−(9/2(t2 ))+6t−5 measured in meters, with t≥0 measured in minutes, and the positive direction meaning forward. a) Find the equation for the velocity at time t. Give units, and find times the times the particle is moving forward. b) Find the equation for the velocity at time t. Give units, and find times the times the particle is moving backwards. c) Find the total distance travelled by the particle. d) Find the equation of the acceleration at time t. Give the units, and find the times when the velocity of the particle is not changing. e) Find the displacement.Consider A runner moving in a positive direction with an initial velocity of 4.6 m/s slows down at a constant rate of -1.3 m/s squared over a period of 2 seconds. What distance does she travel during this process? An object's velocity as a function of time in one dimension is given by the expression; v(t) = 3.8t + 6.68 where are constants have proper SI Units. At what time is the object's velocity 61.2 m/s?
- Dr. John Paul Stapp was a U.S. Air Force officer who studied the effects of extreme acceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating from rest to a top speed of 282 m/s (1015 km/h) in 5.00 s and was brought jarringly back to rest in only 1.40 s. Calculate his (a) acceleration in his direction of motion and (b) acceleration opposite to his direction of motion. Express each in multiples of g (9.80 m/s2) by taking its ratio to the acceleration of gravity.Consider a dragster that, when the light turns green, accelerates uniformly from rest and completes a 372 meter race in a time of 6.61 seconds. a) During this period, what is the dragster's acceleration expressed as a multiple of the acceleration due to gravity, g? b) If the dragster could continue with this average acceleration, what would its speed be, in miles per hour, after it has traveled a total distance of 1.6km1.6km, which is approximately one mile?The velocity of a particle in reference frame A is (5.0î + 5.0ĵ) m/s. The velocity of reference frame A with respect to reference frame B is 6.0 m/s, and the velocity of reference frame B with respect to C is 4.0ĵ m/s. What is the velocity of the particle (in m/s) in reference frame C? (Express your answer in vector form.)
- The figure shows two railway cars with a buffer spring. We want to investigate the transfer of momentum that occurs after car 1 with initial velocity v0 impacts car 2 at rest. The differential equation is given below. Show that the eigenvalues of the coefficient matrix A are λ1=0 and λ2=−c1−c2, with associated eigenvectors v1= 1 1 T and v2= c1 −c2 T. x′′= −c1 c1 c2 −c2 x with ci=k /mi for i=1, 2 The coefficient matrix A is .A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. f(t) = 0.01t4 − 0.03t3 (a) Find the velocity at time t (in ft/s).v(t) = (b) What is the velocity after 1 second(s)?v(1) = ft/s(c) When is the particle at rest? t = s (smaller value) t = s (larger value) (d) When is the particle moving in the positive direction? (Enter your answer using interval notation.) (e) Find the total distance traveled during the first 11 seconds. (Round your answer to two decimal places.) ft(f) Find the acceleration at time t (in ft/s2). a(t) = Find the acceleration after 1 second(s). a(1) = ft/s2 (g) Graph the position, velocity, and acceleration functions for the first 11 seconds. (h) When, for 0 ≤ t < ∞, is the particle speeding up? (Enter your answer using interval notation.) When, for 0 ≤ t < ∞, is it slowing down? (Enter your answer using interval notation.)A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. f(t) = t3 − 15t2 + 72t (a) Find the velocity at time t. v(t) = (b) What is the velocity after 5 s? v(5) = ft/s (c) When is the particle at rest? t = s (smaller value) t = s (larger value) (d) When is the particle moving in the positive direction? (Enter your answer in interval notation.) t (e) Find the total distance traveled during the first 8 s.ft(f) Find the acceleration at time t. a(t) = Find the acceleration after 5 s. a(5) = ft/s2 (g) Graph the position, velocity, and acceleration functions for the first 8 s. (h) When, for 0 ≤ t < ∞, is the particle speeding up? (Enter your answer in interval notation.) When, for 0 ≤ t < ∞, is it slowing down? (Enter your answer in interval notation.)
- In his 1865 science fiction novel From the Earth to the Moon, Jules Verne described how three astronauts are shot to the Moon by means of a huge gun. According to Verne, the aluminum capsule containing the astronauts is accelerated by ignition of nitrocellulose to a speed of 11 km/s along the gun barrel’s length of 220 m. (a) In g units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts? A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston along the gun’s tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves 3.5 km and reaches a speed of 7.0 km/s. Once launched, the rocket can be fired to gain additional speed. (c) In g units, what would be the average acceleration of the rocket…Two masses hanging side by side from springs have positionss1 = 2 sin t and s2 = sin 2t, respectively. At what times in the interval 0 < t do the masses pass eachother?If A = 3i + 3j and B = 7i + 5j, what is the magnitude of C = A - B