Finding unknown velocity and time from distance and accelerations

[chiraganand]

the problem goes like this
YOu are asked to start from rest, go a distance L=105 m and end up at rest again all in minimum time. Your vehicle has a maximum acceleration of a1 and maximum braking of a2. Determint t1, t2 and maximum speed. a1=7.5m/s^2 , a2=-10 m/s^2. The plot of acceleration versus time is attached

I have attempted the solution using the veocity equations but i have 3 unknows the velocity and the times. Tried deriving an expression for accelertionand elocity but not getting it. stuck how to go ahead. Please help

 

[Chestermiller]

the problem goes like this
YOu are asked to start from rest, go a distance L=105 m and end up at rest again all in minimum time. Your vehicle has a maximum acceleration of a1 and maximum braking of a2. Determint t1, t2 and maximum speed. a1=7.5m/s^2 , a2=-10 m/s^2. The plot of acceleration versus time is attached

I have attempted the solution using the veocity equations but i have 3 unknows the velocity and the times. Tried deriving an expression for accelertionand elocity but not getting it. stuck how to go ahead. Please help

Show us what you did so far, please.

 

[chiraganand]

till t1 i found the velocity as 7.5t1 and from t1 to t2 the velocity as -10 (t2-t1). I don't know how to go ahead from this

 

[Chestermiller]

till t1 i found the velocity as 7.5t1 and from t1 to t2 the velocity as -10 (t2-t1). I don't know how to go ahead from this

For the case of constant acceleration, do you know the basic algebraic equations for velocity and distance as a function of time, given the initial velocity v₀ and the initial distance x₀? Do you know the relationship between velocity and distance (without time in the equation) for the case of constant acceleration? Please write these equations down for us. These will be the equations that you will be working with.

Chet

 

[chiraganand]

for constant acceleration, velocity and time is given by v2=v1+at and velocity and distance is given by v2^2=v1^2+2as where s is displacement

 

[Chestermiller]

for constant acceleration, velocity and time is given by v2=v1+at and velocity and distance is given by v2^2=v1^2+2as where s is displacement

Good. Ok. The one other equation I'd like to see is distance versus time. Question before we begin: Is that 105 m the distance covered during t1, or is it the total distance covered from start to stop?

Chet

 

[chiraganand]

Well that is what I qm confused about... because what I have posted is exactly the question that I have. .. not able to figure out if its total distance or not

 

[Chestermiller]

Well that is what I qm confused about... because what I have posted is exactly the question that I have. .. not able to figure out if its total distance or not

Of course, it wouldn't be too complicated to do it both ways. Let's first try the version where it's the total distance. Let x = distance traveled during t1 and let (105 - x) = distance traveled during t2.

Let's first focus on the time interval t1. If a₁ is the acceleration during this time interval, how long does it take (t1) for the vehicle to travel from rest at x = 0 to the location x? What is it's velocity when it reaches x?

Chet

 

[chiraganand]

Velocity at t1 will be either in terms of t1 or distance x
V(t1)^2=2a1x
V(t1)=a1t1

 

[Chestermiller]

Velocity at t1 will be either in terms of t1 or distance x
V(t1)^2=2a1x
V(t1)=a1t1

What is t1 in terms of a and x?
What is V(t1) in terms of a and x?

 

[chiraganand]

T1 wud be sqrt of x/a1

 

[Chestermiller]

T1 wud be sqrt of x/a1

Actually, it would be
t_1=\sqrt{\frac{2x}{a_1}}
The velocity at the end of t₁ would be
v(t_1)=a_1t_1=\sqrt{2a_1x}
Does this make sense so far? If so, we are ready to start the time interval t₂.

 

[chiraganand]

Yep it does... fot the interval t2 the time will be t2-t1 right

 

[Chestermiller]

Yep it does... fot the interval t2 the time will be t2-t1 right

Not really. We are starting the second half of the problem, and we need to use the equation:
s=s_0+v_0t+\frac{1}{2}a_2t^2
For this time interval, s₀=x and v_0=\sqrt{2a_1t_1} where we restart the clock at the end of t1 to t = 0. Also, the velocity during this time interval is given by: v=v₀+a₂t

So, when t = t₂, v = 0 and s = 105. Substitute these values into the above equations and solve for t₂ and then x.

 

[chiraganand]

But I am getting two unknows now.. t2 and x

 

[Chestermiller]

But I am getting two unknows now.. t2 and x

Yes, but you have two equations. Solve the velocity equation for t2 in terms of x, and then substitute that into the displacement equation.

Chet

 

[chiraganand]

Still not able to get it...seems to be easy but not getting it

 

[Chestermiller]

Still not able to get it...seems to be easy but not getting it

v_0=\sqrt{2a_1t_1} and 0 = v₀+a₂t₂.

1. Solve for t₂
2. Substitute into 105 = v₀t₂+0.5a₂(t₂)²
3. Solve for x

 

[chiraganand]

But we still have t1 as unkown

 

[chiraganand]

Got the answer.. thank you for your help