Calculus Help Please

Question

What constant acceleration is required to increase the speed of a car from 29 mi/h to 54 mi/h in 3 s? (Round your answer to two decimal places.)

 ft/s²

Answer 1

Hi Angie!

So with any math problem. I like to ask what we are looking for? How can we write what the have given us in math terms?

Well first, the problem wants us to find the acceleration required to do something, lets call that "a" and know that is constant.

We know the car starts with an initial velocity of 29 mi/h ('v_i") ends with a speed of 54 mi/h ('v_f"), and does so from t_i=0s to t_f=3s.

Now that we wrote everything down in math terms, we can use equations. If you have gone through the constant acceleration kinematic equations in class, you can use v_f = v_i + a*(t_f - t_i) (make sure you have correct units on both sides.

But since you are in a calculus class lets try and derive the answer.

we know that by definition acceleration is the time derivative of velocity

a = dv/dt

If we rearrange that get

adt = dv

Integrate both sides

∫a*dt = ∫dv.

Because a is constant

a∫dt = ∫dv

Integrate from initial to final states

a*(t_f - t_i) = v_f - v_i

And notice this is the same as the kinematic equation. I'll let you try and plug in values and solve, but just remember to watch out for units. Hope this helps!

- Evan

Answer 2

Acceleration = a, which is equal to dv/dt (since acceleration is the derivative of velocity)

dv/dt in this case can be represented by Δv/Δt since you are given definite times and speed.

Δt = dt in this case as well.

Δ is always final - initial in this case.

constant acceleration (a) = dv/dt = Δv/Δt = (final velocity - initial velocity)/time

a = (54 mi/h - 29 mi/h)/3 seconds [for units to match up, convert the 3 second into hours by multiplying 3 seconds by conversion factor (1hr/3600s) - this would result in 3 seconds being converted to 0.00083 hours]

a = (25 mi/h)/0.00083 h

a = 30000.012 mi/h²

to get your answer to ft/s² multiply answer by following conversion factors:(5280 ft/1 mi) & (1 h/3600 s)²

30000.012 mi/h² * (5280 ft/1 mi) * (1 h/3600 s)² = 44000.0176 ft/s²