# Finding Velocity of Particle: Problem Solved

• tennistudof09
In summary, the problem is about finding the velocity of a particle at time t=0, given its position function s(t)=2t^3 - 21t^2 + 60t. The derivative is taken to get the velocity function, which is then evaluated at t=0 to get a velocity of 60ft/sec. The correctness of this answer can also be confirmed by looking at the displacement, velocity, and acceleration vectors.

#### tennistudof09

I wanted to make sure I did this problem correctly. The problem is:

A particle moves along a straight line and its position at time t is given by s(t)=2t^3 - 21t^2 + 60t where s is measured in feet and t in seconds.

Find the velocity of the particle when t=0.

I took the derivative of s(t) and got 6t^2 - 42t + 60, and then substituted 0 in for t. I got 60ft/sec for the answer. Is this correct??

$$\frac{ds}{dt}=v(t)$$

$$\frac{dv}{dt}=a(t)$$

Where $$s(t)$$ is the displacement vector, $$v(t)$$ is the velocity vector and
$$a(t)$$ is the acceleration vector.

If you look at it in any given point, so I'm guessing that it's correct.

## 1. How do you calculate the velocity of a particle?

The velocity of a particle can be calculated by dividing the displacement of the particle by the time it took to travel that distance. This can be represented by the equation v = Δx / Δt, where v is the velocity, Δx is the displacement, and Δt is the time.

## 2. What is the difference between average velocity and instantaneous velocity?

Average velocity is the overall displacement of a particle divided by the total time it took to travel that distance. Instantaneous velocity, on the other hand, is the velocity of the particle at a specific moment in time. It is calculated by taking the limit as the time interval approaches zero in the equation v = Δx / Δt.

## 3. How does acceleration affect the velocity of a particle?

Acceleration is the rate of change of velocity over time. If a particle is accelerating, its velocity will either increase or decrease depending on the direction of the acceleration. A positive acceleration will result in an increase in velocity, while a negative acceleration will result in a decrease in velocity.

## 4. Can the velocity of a particle change without it being accelerated?

Yes, the velocity of a particle can change even if it is not being accelerated. This can happen if the particle is moving in a curved path, where the direction of its velocity is constantly changing, but the magnitude of its velocity remains constant. This is known as uniform circular motion.

## 5. How can we use calculus to find the velocity of a particle?

Calculus can be used to find the velocity of a particle by taking the derivative of the position function with respect to time. This will give the instantaneous velocity of the particle at any given time. Integrating the acceleration function with respect to time can also be used to find the velocity of a particle. This is known as the fundamental theorem of calculus.

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