What is the average speed of an object that is not moving at #t=0# and accelerates at a rate of #a(t) =4-t# on #t in [2,3]#?

Question

Isabella Alvarez · Accepted Answer

Remember how one calculates average value of any quantity:
#\langle f \rangle = \frac{1}{t_2-t_1}\int_{t_1}^{t_2} f(t)dt# We need to calculate #\langle v \rangle#, where #v# is the velocity, for that we will need to first find the velocity it self, but we do know that 
#\frac{dv}{dt}=a#, where a is the acceleration or in other words:
#v(t_2)-v(t_1)= \int_{t_1}^{t_2}a(t)dt# We have been given:
#a(t)=4-t# and #v(0)=0#, lets us plug these in the above formula, with #t_1=0#, #t_2=t#
We get,
#v(t)=\int_{0}^{t}(4-t)dt#
#v(t)= (4t-\frac{t^2}{2})#
Now for its average value we use the first formula used
#\langle v \rangle = \frac{1}{3-2}\int_{2}^{3}(4t-\frac{t^2}{2}) dt#
#\langle v \rangle = \frac{1}{3-2}(2t^2-\frac{t^3}{6}) # with limits 2 and 3.
Inserting limits
#\langle v \rangle = (50-19/6) #
#\langle v \rangle = (281/6)=46.8 units/sec #

Nicholas Agrusa · Answer

undefined To find the average speed of an object, we need to calculate the total distance traveled divided by the total time taken. Given that the object starts at rest at t=0 and accelerates at a rate of a(t) = 4 - t on t in [2,3], we can integrate the acceleration function to find the velocity function. Then, we integrate the velocity function over the interval [2,3] to find the total distance traveled. Finally, we divide this distance by the time interval (which is 1 hour) to find the average speed. The velocity function can be found by integrating the acceleration function, which gives v(t) = 4t - (1/2)t^2 + C. Using the initial condition v(2) = 0, we find that C = -4. Therefore, v(t) = 4t - (1/2)t^2 - 4. Integrating v(t) over the interval [2,3], we get the total distance traveled. After finding this distance, we divide it by the time interval (1 hour) to find the average speed.