Physics, 10.12.2019 02:31 fy1lqjh

Aparticle's trajectory is described by x =(12t3−2t2)m and y =(12t2−2t)m, where t is in s.
what is the particle's speed at t=0s ? what is the particle's speed at t=5.0s ? express your answer using two significant figures. what is the particle's direction of motion, measured as an angle from the x-axis, at t=0 s ?
express your answer using two significant figures. what is the particle's direction of motion, measured as an angle from the x-axis, at t=5.0s ?
express your answer using two significant figures.

Answers: 2

Show answers

Answers

Answer from: nmooesy2229

2 m/s890 m/s270°7.6°Further explanation

Given:

A particle's trajectory is described by

$\boxed{ \ x = (12t^3 - 2t^2) \ m \ }, and$ $\boxed{ \ y = (12t^2 - 2t) \ m \ }$

We can see that the particle moves in the direction in both the x-axis and the y-axis.

The instantaneous velocity component is obtained from the first derivative of the position function with respect to time.

$\boxed{ \ v_x = \frac{dx}{dt} \ } \ and \ \boxed{ \ v_y = \frac{dy}{dt} \ }$

Let's arrange the velocity function for horizontal and vertical components.

$\boxed{ \ v_x = 36t^2 - 4t \ m/s \ }$ $\boxed{ \ v_y = 24t - 2 \ m/s \ }$

If asked the vector of the velocity, then $\boxed{ \ \vec{v} = (36t^2 - 4t) \hat{i} + (24t - 2) \hat{j} \ m/s \ }$

- - - - - - -

Determine the particle's speed at t = 0 s.

$t = 0 \rightarrow \boxed{ \ v_x = 36(0)^2 - 4(0) \ } \rightarrow \boxed{ \ v_x = 0 \ m/s \ }$

$t = 0 \rightarrow \boxed{ \ v_y = 24(0) - 2 \ } \rightarrow \boxed{ \ v_y = -2 \ m/s \ }$

In vector form, we can write it as $\boxed{ \ \vec{v} = - 2 \hat{j} \ m/s \ }$

From the two velocity components above, we calculate the result.

$\boxed{ \ v = \sqrt{v_x^2 + v^2_y} \ } \rightarrow \boxed{ \ v = \sqrt{0^2 + (-2)^2} \ }$

Hence, the speed of the particle at t = 0 is given by $\boxed{ \ v = 2 \ m/s \ }$

Determine the particle's speed at t = 5 s.

$t = 5 \rightarrow \boxed{ \ v_x = 36(5)^2 - 4(5) \ } \rightarrow \boxed{ \ v_x = 880 \ m/s \ }$

$t = 5 \rightarrow \boxed{ \ v_y = 24(5) - 2 \ } \rightarrow \boxed{ \ v_y = 118 \ m/s \ }$

In vector form, we can write it as $\boxed{ \ \vec{v} = 880 \hat{i} + 118 \hat{j} \ m/s \ }$

From the two velocity components above, we calculate the result.

$\boxed{ \ v = \sqrt{v_x^2 + v^2_y} \ } \rightarrow \boxed{ \ v = \sqrt{880^2 + 118^2} = 887.8761175 \ }$

Hence, the speed of the particle at t = 5 is given by $\boxed{ \ v = 890 \ m/s \ }$ in two significant figures.

- - - - - - -

Now we determine the particle's direction of motion, measured as an angle from the x-axis, at t = 0 s and t = 5 s.

The direction of motion of a particle is given by an α angle which can be calculated by the trigonometric formula.

$\boxed{ \ tan \ \alpha = \frac{v_y}{v_x} \ } \rightarrow \boxed{ \ \alpha = arctan\frac{v_y}{v_x} \ }$

At t = 0 s,

$\boxed{ \ v_x = 0 \ } \ m/s$ $\boxed{ \ v_y = -2 \ } \ m/s$

$\boxed{ \ \alpha = arctan\frac{-2}{0} \ } \rightarrow \boxed{ \ \alpha = arctan( - \infty ) \ }$

Actually, from the two components of velocity, we can see the particle's direction of motion.

$\boxed{ \ v_x = 0 \ } \ m/s$ → it has no component. $\boxed{ \ v_y = -2 \ } \ m/s$ → move along in the negative y-axis direction.

Therefore the particle's direction of motion at t = 0 s is given by the angle $\boxed{ \ \alpha = 270^0 \ }$

At t = 5 s,

$\boxed{ \ v_x = 880 \ } \ m/s$ $\boxed{ \ v_y = 118 \ } \ m/s$

$\boxed{ \ \alpha = arctan\frac{118}{880} \ } \rightarrow \boxed{ \ \alpha = 7.6^0 \ }$ in two significant figures.

Therefore the particle's direction of motion at t = 5 s is given by the angle $\boxed{ \ \alpha = 7.6^0 \ }$

Learn moreThe theoretical density of platinum which has the FCC crystal structure. Determine the acceleration of the bear The energy density of the stored energy

Keywords: a particle's trajectory, speed, express your answer using two significant figures, the direction of motion, derivative, the instantaneous velocity component, horizontal, vertical, vector, the angle

Aparticle's trajectory is described by x =(12t3−2t2)m and y =(12t2−2t)m, where t is in s. what is th

Answer from: Quest

Answer59.2kg; /

Answer from: Quest

Wut

Another question on Physics

Physics, 22.06.2019 12:50

Arunner is jogging at a steady 3.6 km/hr. when the runner is 2.9 km from the finish line, a bird begins flying from the runner to the finish line at 14.4 km/hr (4 times as fast as the runner). when the bird reaches the finish line, it turns around and flies back to the runner. even though the bird is a dodo, we will assume that it occupies only one point in space, i.e., a zero-length bird. how far does the bird travel? (b) after this first encounter, the bird then turns around and flies from the runner back to the finish line, turns around again and flies back to the runner. the bird repeats the back and forth trips until the runner reaches the finish line. how far does the bird travel from the beginning? (i.e. include the distance traveled to the first encounter)

Answers: 2

Answer

Physics, 22.06.2019 15:30

What can you increase the strength of an electromagnet by

Answers: 2

Answer

Physics, 22.06.2019 18:00

Astudent weighing 700 n climbs at constant speed to the top of an 8 m vertical rope in 10 s. the average power expended by the student to overcome gravity is most nearly

Answers: 2

Answer

Physics, 22.06.2019 21:00

The amount of work done is determined by 2 factors. describe an example of work (using the scientific definition) list the 2 factors that determine the amount of work done.

Answers: 2

Answer

You know the right answer?

Aparticle's trajectory is described by x =(12t3−2t2)m and y =(12t2−2t)m, where t is in s.
what...

Questions

Physics, 22.01.2021 06:00

An 85 kg log falls off a 27.2 m cliff. Calculate the KE the log will have just before it hits the ground. Assume there is no friction or air resistanc...