A particle with velocity at any time t is given by v(t)=2e^{2t} moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

A particle with velocity at any time t is given by v(t)=2e^{2t} moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

asked 2021-03-07
A particle with velocity at any time t is given by \(\displaystyle{v}{\left({t}\right)}={2}{e}^{{{2}{t}}}\) moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

Answers (1)

Given that the velocity of a particle at any time t is given by
and it is moving in the straight line we have to find the displacement. Integrate the above Equation between the limits 2 to 4

Relevant Questions

asked 2021-02-19
A 10 kg objectexperiences a horizontal force which causes it to accelerate at 5 \(\displaystyle\frac{{m}}{{s}^{{2}}}\), moving it a distance of 20 m, horizontally.How much work is done by the force?
A ball is connected to a rope and swung around in uniform circular motion.The tension in the rope is measured at 10 N and the radius of thecircle is 1 m. How much work is done in one revolution around the circle?
A 10 kg weight issuspended in the air by a strong cable. How much work is done, perunit time, in suspending the weight?
A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total workdone on the block over the same distance?
What is the kinetic energy of a 2 kg ball that travels a distance of 50 metersin 5 seconds?
A ball is thrown vertically with a velocity of 25 m/s. How high does it go? What is its velocity when it reaches a height of 25 m?
A ball with enough speed can complete a vertical loop. With what speed must the ballenter the loop to complete a 2 m loop? (Keep in mind that the velocity of the ball is not constant throughout the loop).
asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors \(\displaystyle\hat{{{i}}}\) and \(\displaystyle\hat{{{j}}}\).
1. (Enter in box 1) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+\)
5. ( Enter in box 5) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
asked 2021-04-18
A 2.0-kg projectile is fired with initial velocity components \(\displaystyle{v}_{{{0}{x}}}={30}\) m/s and \(\displaystyle{v}_{{{0}{y}}}={40}\) m/s from a point on the earth's surface. Neglect any effects due to air resistance. What is the kinetic energy of the projectile when it reaches the highest point in its trajectory? How much work was done in firing the projectile?
asked 2021-03-24
A 2.4-kg object is attached to a horizontal spring of forceconstant k=4.5 kN/m. The spring is stretched 10 cm fromequilibrium and released. Find (a) the frequency of themotion, (b) the period, (c) the amplitude, (d) the maximum speed,and (e) the maximum acceleration. (f) When does the objectfirst reach its equilibrium position? What is itsacceleration at this time?
Two identical blocks placed one on top of the other rest on africtionless horizontal air track. The lower block isattached to a spring of spring constant k= 600 N/m. Whendisplaced slightly from its equilibrium position, the systemoscillates with a frequency of 1.8 Hz. When the amplitude ofoscillation exceeds 5 cm, the upper block starts to slide relativeto the lower one. (a) What are the masses of the twoblocks? (b) What is the coefficient of static frictionbetween the two blocks?
asked 2021-05-09
A pair of speakers separated by .700 m are driven by the sameoscillator at a frequency of 690 Hz. An observer originallypositioned at one of the speakers begins to walk along a lineperpendicular to the line joining the speakers.
a. How far must the observer walk before reaching a relativemaximum in intensity?
b. How far will the observer be from the speaker when the firstrelative minimum is detected in the intensity?
asked 2021-03-22
A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant but starts at .100 at P and increases linerly with distance past P, reaching a value of .600 at 12.5 m past point P. (a) Use the work energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid iff the friciton coefficient didn't increase, but instead had the constant value of .1?
asked 2021-03-12
A 75.0-kg man steps off a platform 3.10 m above the ground. Hekeeps his legs straight as he falls, but at the moment his feettouch the ground his knees begin to bend, and, treated as aparticle, he moves an additional 0.60 m before coming torest.
a) what is the speed at the instant his feet touch theground?
b) treating him as a particle, what is his acceleration(magnitude and direction) as he slows down, if the acceleration isassumed to be constant?
c) draw his free-body diagram (see section 4.6). in termsof forces on the diagram, what is the net force on him? usenewton's laws and the results of part (b) to calculate the averageforce his feet exert on the ground while he slows down. expressthis force in newtons and also as a multiple of his weight.
asked 2021-05-09
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.
Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.
Part B:
A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
asked 2021-05-10
Hypothetical potential energy curve for aparticle of mass m
If the particle is released from rest at position r0, its speed atposition 2r0, is most nearly
a) \(\displaystyle{\left({\frac{{{8}{U}{o}}}{{{m}}}}\right)}^{{1}}{\left\lbrace/{2}\right\rbrace}\)
b) \(\displaystyle{\left({\frac{{{6}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}\)
c) \(\displaystyle{\left({\frac{{{4}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}\)
d) \(\displaystyle{\left({\frac{{{2}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}\)
e) \(\displaystyle{\left({\frac{{{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}\)
if the potential energy function is given by
where b and c are constants
which of the following is an edxpression of the force on theparticle?
1) \(\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left({r}^{{-\frac{{5}}{{2}}}}\right)}\)
2) \(\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{3}{b}\right\rbrace}{\left\lbrace{2}\right\rbrace}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}\)
3) \(\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{3}\right\rbrace}{\left\lbrace{2}\right\rbrace}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}\)
4) \(\displaystyle{2}{b}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}+{c}{r}\)
5) \(\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{2}{b}\right\rbrace}{\left\lbrace{5}\right\rbrace}{\left({r}^{{-\frac{{5}}{{2}}}}\right)}+{c}{r}\)
asked 2021-02-19
An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of \(\displaystyle{1950}\ {N}\cdot{m}\) to the propeller, which starts from rest.
a) What is the angular acceleration of the propeller? Model the propeller as a slender rod.
b) What is the propeller's angular speed after making 5.00 revolutions?
c) How much work is done by the engine during the first 5.00 revolutions?
e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?