In this question, we intend to study the motion
of a particle (M) in three phases:
1. Motion along BC
The particle (M) is launched horizontally
at to = 0 on a surface BC with an initial
speed Vo = Vb = 15 m/s.
The speed of (M) decreases until it becomes
zero at a point C.
The time equation of (M) along x -axis oriented positively from B to C is:
x= -6.25 t² + 15 t S.I.
Determine the length of BC and the time t1 needed by M to cover BC.
2. Motion along CD
(M) continues down the straight track CD at an acceleration of 3 m/s². The particle needs a duration
t2 = 2s to reach the point D.
Determine the speed of the particle at the point D, and deduce the distance CD.
3. Motion along DE
The particle continues along the path DE. The time equation of (M) along the x-axis oriented positively
from D to E is: x = 6t S.I.
Determine the time t3 needed by (M) to cover the distance DE = 6 m along this path.
4. The whole jouney
Determine the average speed of the whole motion of (M) along the track BCDE.

Question

toniand toniand

02-02-2024
Physics

contestada

In this question, we intend to study the motion
of a particle (M) in three phases:
1. Motion along BC
The particle (M) is launched horizontally
at to = 0 on a surface BC with an initial
speed Vo = Vb = 15 m/s.
The speed of (M) decreases until it becomes
zero at a point C.
The time equation of (M) along x -axis oriented positively from B to C is:
x= -6.25 t² + 15 t S.I.
Determine the length of BC and the time t1 needed by M to cover BC.
2. Motion along CD
(M) continues down the straight track CD at an acceleration of 3 m/s². The particle needs a duration
t2 = 2s to reach the point D.
Determine the speed of the particle at the point D, and deduce the distance CD.
3. Motion along DE
The particle continues along the path DE. The time equation of (M) along the x-axis oriented positively
from D to E is: x = 6t S.I.
Determine the time t3 needed by (M) to cover the distance DE = 6 m along this path.
4. The whole jouney
Determine the average speed of the whole motion of (M) along the track BCDE.

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masifakrami masifakrami · Answer 1 · 2024-02-03T05:05:19+01:00

Answer:

1) The length of BC is 0 meters.

2) The speed of the particle at point D is 6 m/s.
3) Therefore, the time t3 needed by M to cover the distance DE is 1 second.

4) since we do not have the distance CD, we cannot calculate the average speed accurately.

Explanation:

1. Motion along BC:

The length of BC can be determined by finding the time t1 needed for particle M to cover BC. The equation given for the motion of M along the x-axis is x = -6.25t² + 15t.

To find t1, we set x = 0 since the speed of M becomes zero at point C. We solve the equation -6.25t² + 15t = 0 for t.

By factoring out t, we get t(-6.25t + 15) = 0.

This gives us two possible solutions: t = 0 or -6.25t + 15 = 0.

Since time cannot be negative, we discard the t = 0 solution.

To solve -6.25t + 15 = 0, we isolate t:

-6.25t = -15

Dividing both sides by -6.25, we find t = 15 / 6.25 = 2.4 seconds.

Therefore, the time t1 needed by M to cover BC is 2.4 seconds.

The length of BC is given by substituting t = t1 into the equation x = -6.25t² + 15t:

x = -6.25(2.4)² + 15(2.4)

Simplifying the equation, we find x = 0.

Hence, the length of BC is 0 meters.

2. Motion along CD:

The article continues down the straight track CD with an acceleration of 3 m/s². The duration t2 needed to reach point D is given as 2 seconds.

Using the equation of motion: x = x0 + v0t + 0.5at², where x0 is the initial position, v0 is the initial velocity, a is the acceleration, and t is the time.

Since the particle starts at rest (v0 = 0) and the initial position is not given (x0 = ?), we cannot calculate the distance CD directly.

However, we can find the speed of the particle at point D using the equation v = v0 + at, where v is the final velocity, v0 is the initial velocity, a is the acceleration, and t is the time:

v = 0 + (3)(2)

Simplifying the equation, we find v = 6 m/s.

Hence, the speed of the particle at point D is 6 m/s.

3. Motion along DE:

The time equation of M along the x-axis, oriented positively from D to E, is given as x = 6t.

To determine the time t3 needed by M to cover the distance DE = 6 meters, we substitute x = DE = 6 into the equation:

6 = 6t

Simplifying the equation, we find t = 1 second.

Therefore, the time t3 needed by M to cover the distance DE is 1 second.

4. The whole journey:

To find the average speed of the whole motion of M along the track BCDE, we need to calculate the total distance traveled and the total time taken.

The total distance traveled is the sum of the distances BC, CD, and DE, which is 0 +. + 6 = ? meters.

The total time taken is the sum of the times t1, t2, and t3, which is 2.4 + 2 + 1 = 5.4 seconds.

To find the average speed, we divide the total distance by the total time:

average speed = total distance / total time

However, since we do not have the distance CD, we cannot calculate the average speed accurately.

In conclusion, we can determine the length of BC as 0 meters and the time t1 needed by M to cover BC as 2.4 seconds. We know that the speed of the particle at point D is 6 m/s. The time t3 needed by M to cover the distance DE is 1 second. However, we cannot determine the average speed of the whole motion without the distance CD.