A body moves in a straight line,with initial
velocity and uniform acceleration.find how
far it travels in 12sec.,given that it travels
246m in the first 6 sec.,and 69m in the last
3sec.find the initial velocity.
Please show step
s(t) = s0 + v0t + a/2 t^2
so,
s0 + 6v0 + 18a = 246
s0 + 9v0 + 81/2 a = 246+69
Solve for v0
6 V0 + 18 a = 246
V0 + 9 a = V1
3 V1 + 4.5 a = 69
subs ... 3 V0 + 27 a + 4.5 a = 69
... times 2 ... 6 V0 + 63 a = 138
subtract to eliminate V0
... 45 a = -108
... a = -2.4
substitute back to find V0
Step 1: Understand the problem
We are given that a body moves in a straight line with an initial velocity and uniform (constant) acceleration. We need to find the distance it travels in 12 seconds and also determine the initial velocity. Additionally, it is mentioned that the body travels 246m in the first 6 seconds and 69m in the last 3 seconds.
Step 2: Use the equations of motion
We can solve this problem using the equations of motion under constant acceleration. The equations we need to use are:
1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as
Where:
- s is the displacement (distance) covered by the body
- u is the initial velocity
- v is the final velocity
- a is the acceleration
- t is the time taken
Step 3: Find the acceleration
Given that the body has uniform acceleration, we can find the acceleration using the formula:
a = (v - u) / t
For the first 6 seconds:
v = ?
u = ?
t = 6 seconds
s = 246m
Using equation 2, we have:
s = ut + (1/2)at^2
246 = u(6) + (1/2)a(6)^2
246 = 6u + 18a
For the last 3 seconds:
v = ?
u = ?
t = 3 seconds
s = 69m
Using equation 2 again, we have:
s = ut + (1/2)at^2
69 = u(3) + (1/2)a(3)^2
69 = 3u + (9/2)a
Step 4: Solve the equations
Now we have a system of two equations with two unknowns (u and a):
246 = 6u + 18a ...(Equation 1)
69 = 3u + (9/2)a ...(Equation 2)
First, multiply Equation 2 by 2 to eliminate fractions:
138 = 6u + 9a ...(Equation 2)
Next, subtract Equation 1 from Equation 2:
(138 - 246) = (6u + 9a) - (6u + 18a)
-108 = -9a
a = 12 m/s^2
Step 5: Find the initial velocity
Using Equation 1 and the value of acceleration (a = 12 m/s^2), we can find u:
246 = 6u + 18(12)
246 = 6u + 216
6u = 30
u = 5 m/s
Step 6: Find the distance traveled in 12 seconds
To find the distance traveled in 12 seconds, we can use Equation 2:
s = ut + (1/2)at^2
s = (5)(12) + (1/2)(12)(12)
s = 60 + 72
s = 132
Therefore, the body travels 132 meters in 12 seconds and the initial velocity (u) is 5 m/s.