An automobile starts from rest and accelerates to a final velocity in two stages along a straight road. Each stage occupies the same amount of time. In stage 1, the magnitude of the car's acceleration is 2.7 m/s2. The magnitude of the car's velocity at the end of stage 2 is 2.8 times greater than it is at the end of stage 1. Find the magnitude of the acceleration in stage 2.
time for each stage = T
V1 = 2.7 T
V2 = 2.7 T + a T
2.7 T + a T = 2.8 (2.7 T)
2.7 + a = 2.8 * 2.7
a = 1.8*2.7
7.56
To find the magnitude of the acceleration in stage 2, we can start by considering the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
In stage 1, the car starts from rest, so the initial velocity (u1) is 0. The magnitude of the acceleration (a1) is given as 2.7 m/s^2. Let's denote the final velocity at the end of stage 1 as v1.
Therefore, the equation for stage 1 becomes:
v1^2 = 0 + 2 * (2.7) * s1
v1^2 = 5.4s1 (Equation 1)
In stage 2, the car continues from the end of stage 1 with a new acceleration (a2). Given that the final velocity at the end of stage 2 (v2) is 2.8 times greater than v1, we can express it as:
v2 = 2.8 * v1
We can use this information along with the equation of motion to find the acceleration for stage 2.
Plugging in the values into the equation of motion for stage 2:
v2^2 = v1^2 + 2 * a2 * s2
(2.8 * v1)^2 = v1^2 + 2 * a2 * s2
7.84 * v1^2 = v1^2 + 2 * a2 * s2 (Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with two variables (v1 and a2). We can solve them simultaneously to find the values.
First, let's solve Equation 1 for s1:
s1 = v1^2 / 5.4
Next, substitute this value into Equation 2:
7.84 * v1^2 = v1^2 + 2 * a2 * (v1^2 / 5.4)
Now, simplify the equation:
7.84 * v1^2 = v1^2 + (2/5.4) * a2 * v1^2
7.84 = 1 + (2/5.4) * a2
Rearrange the equation to isolate a2:
(2/5.4) * a2 = 7.84 - 1
(2/5.4) * a2 = 6.84
Finally, solve for a2:
a2 = (6.84 * 5.4) / 2
a2 = 18.48 m/s^2
Therefore, the magnitude of the acceleration in stage 2 is 18.48 m/s^2.