# Mechanics- general motion in a straight line.

• MHB
• Shah 72
In summary, the velocity of a particle moving in a straight line is given by v = -t^3 + 9t m/s for 0 < t < 5. When t = 5s, the displacement of the particle from its original position is -43.8m. To find the distance traveled by the particle from t = 0 to t = 5, we can use the formula D = ∫|v(t)|dt or calculate it in two parts by integrating from 0 to 3 and from 3 to 5. The result in both cases is 84.3m.
Shah 72
MHB
A particle moves in a straight line. The velocity of the particle, v m/s, at time t s is given by v= -t^3+9t m/s for 0<t<5
a) Find the displacement of the particle from its original position, when t=5s
I got the ans for this by integration and limits 5 and 0 =- 43.8
b) work out the distance that the particle travels from t= 0 to t=5
I don't understand this. Velocity is positive from 0 to 3 and negative from 3 to 5 when I plot the velocity time graph.
I tried integration again with limits 3 to 0 and the next limit from 5 to 3. Iam not getting the ans which is 84.3m

in general, distance traveled is the integral of speed …

$\displaystyle D = \int_{t_0}^{t_f} |v(t)| \, dt$

Note the velocity in this problem is positive in the interval (0,3) and negative in the interval (3,5]

two ways to do this …

$\displaystyle D = \int_0^3 9t-t^3 \, dt + \int_3^5 t^3 - 9t \, dt$

$\displaystyle D = \int_0^3 9t-t^3 \, dt - \int_3^5 9t-t^3 \, dt$

skeeter said:
in general, distance traveled is the integral of speed …

$\displaystyle D = \int_{t_0}^{t_f} |v(t)| \, dt$

Note the velocity in this problem is positive in the interval (0,3) and negative in the interval (3,5]

two ways to do this …

$\displaystyle D = \int_0^3 9t-t^3 \, dt + \int_3^5 t^3 - 9t \, dt$

$\displaystyle D = \int_0^3 9t-t^3 \, dt - \int_3^5 9t-t^3 \, dt$
Thank you very much!

## What is mechanics?

Mechanics is a branch of physics that deals with the behavior of physical bodies when subjected to forces or displacements, and the subsequent motion and effects of those bodies.

## What is general motion in a straight line?

General motion in a straight line is the movement of an object along a single path, without any change in direction or deviation from that path.

## What are the basic principles of mechanics?

The basic principles of mechanics include Newton's laws of motion, the principle of conservation of energy, and the principle of conservation of momentum.

## What is the difference between speed and velocity?

Speed is a measure of how fast an object is moving, while velocity is a measure of how fast an object is moving in a specific direction. In other words, velocity takes into account both the speed and direction of an object's motion.

## What factors affect an object's motion in a straight line?

The factors that affect an object's motion in a straight line include the magnitude and direction of the applied force, the mass of the object, and any external forces acting on the object (such as friction or air resistance).

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