1)A car is moving with uniform accelaration. It moves along two point A and B and enters with velocity 50 m/s and 80 m/s respectively . Find the velocity when the car is moving in b/w the point A and B.
Let t be the time measured from when the velocity is 50 m/s
V = 50 + a t
If T is the time it takes to go from 50 to 80 m/s,
V(t) = 50 + 30 t/T
(The acceleration is a = 30/T)
Now integrate for X(t)
X(t) = 50 t + 15 t^2/T
X(T) = 50 T + 15 T = 65 T (point B location)
X(0) = 0 (point A location)
You want the time when X = X(T)/2
50 t + 15 t^2/T = 32.5 T
50 (t/T) + 15(t/T)^2 = 32.5
Solve for t/T
t/T = 0.556
V at that time is 50 + 0.556*30 = 66.7 m/s.. a bit more than the average speed
For the question asked above this was the solution given but as I am not introduced to the integration concept is there any other method of solving this problem with the concepts I know that a class 9 students know.
The equation
X(t) = 50 t + 15 t^2/T
does not have to be derived with calculus.
It is just the familiar equation
X = Vo*t + (a/2)*t^2
In your case, Vo = 50 and a = (V2 - V1)/T
The rest of the derivation can proceed without calculus.
Yes, there is another method to solve this problem without using integration. Here's how you can approach it:
1. First, calculate the time it takes for the car to go from 50 m/s to 80 m/s using the formula:
t = (Vf - Vi) / a, where Vf is the final velocity (80 m/s), Vi is the initial velocity (50 m/s), and a is the acceleration.
2. Once you have the value of time (t), you need to find the distance covered by the car during this time. To do this, use the formula:
d = Vi*t + (1/2)*a*t^2, where d is the distance covered, Vi is the initial velocity (50 m/s), a is the acceleration, and t is the time calculated in step 1.
3. Now, divide the distance covered (d) by 2 to find the midpoint between points A and B.
4. To find the velocity at this midpoint, calculate the time it takes for the car to cover half the distance using the formula:
t' = sqrt((2*d) / a), where d is the distance calculated in step 3, and a is the acceleration.
5. Finally, use the formula:
V = Vi + a * t', where Vi is the initial velocity (50 m/s), a is the acceleration, and t' is the time calculated in step 4.
By following these steps, you can find the velocity of the car when it is moving between points A and B without using integration.