Velocity and Acceleration within first 10 seconds

[Perdition]

Homework Statement

Body's linear equation of motion is s= A + B*t + C*t^3 (A= 2.0m, B = 3.0 m/s, C = 0.010 m/s^3) What is the velocity and acceleration of a body on t= 0 s and t = 10 s. Calculate the velocity and accelerations average values within first 10 seconds, starting from start of the movement.

The Attempt at a Solution

For the first part of problem, i took derivative of equation to get velocity and 2nd derivative for acceleration

s' = B + 3*C*t^2
s'' = 6*C*t

and calculated the valuesNow for the second part, what i tried to do is replace t within the velocity equation with t=0 added by t= 1 added by t=2 and so forth all the way to 10 and divide the result by 10. I was wondering if this is the right way do this and if I am correct is there a shorter way?

Thanks in advance

 

[TSny]

Hi perdition. Welcome to PF.

The way you are calculating the average velocity by finding the velocity at each second and averaging these velocities will only give you an approximation for the true average velocity.

Most textbooks will give you the definition for average velocity as the total displacement divided by the time interval: v_avg = Δs/Δt.

There's a similar formula for finding average acceleration.

Your method would approach the true average if you take smaller time intervals. For example, if you found the velocity at every tenth of a second and divided the sum of these velocities by the number of velocities (100) you would get a closer value to the true average velocity. Doing it for every hundredth of a second would be even better. But it's much easier and you get the exact answer by just using Δs/Δt.

 

[Perdition]

English isn't my first language, can you confirm if i understood it correctly please.

I use the normal v= s/t formula. I get value of s by placing t=10 to the equation of motion given to me? The value s then I divide by 10?
edit: oh okey i got it now. Thanks a lot :)

 

[TSny]

I think you have it. Just to make sure: v_avg = Δs/Δt = (s_f-s_i)/(t_f-t_i)

t_i = 0, t_f = 10

s_i = the value of s at t = t_i, s_f = the value of s at t = t_f

 

[Nickie]

I would say that your approach is correct. To calculate the average velocity and acceleration within the first 10 seconds, you can use the average value theorem, which states that the average value of a function over an interval is equal to the function's value at the midpoint of that interval. In this case, the interval is from t=0 to t=10, so the midpoint is t=5. This means that the average velocity and acceleration can be calculated by plugging t=5 into the velocity and acceleration equations. However, your approach of dividing the sum of the values at different time points by the number of time points is also valid and may be easier to use in certain situations. Ultimately, both methods will give you the same result and it is up to you to choose which one is more convenient for your specific problem.