A particle is travelling on a straight line with constant acceleration. It covers 6m in the 4th second and 7m in the 5th second. Fine its acceleration and initial speed.
A=7÷4
To find the acceleration and initial speed of the particle, we can use the equations of motion for uniformly accelerated linear motion.
The equations of motion are:
1. Displacement (s) = initial velocity (u) × time (t) + 0.5 × acceleration (a) × time squared (t^2)
2. Final velocity (v) = initial velocity (u) + acceleration (a) × time (t)
3. Displacement (s) = (final velocity (v)^2 - initial velocity (u)^2) / (2 × acceleration (a))
Let's solve the problem step-by-step.
Step 1: Find the acceleration (a)
Given:
- Displacement in the 4th second = 6m
- Displacement in the 5th second = 7m
Using the first equation of motion, we can calculate the displacements s4 and s5:
s4 = u × 4 + 0.5 × a × 4^2
s5 = u × 5 + 0.5 × a × 5^2
Substituting the values we know:
6 = 4u + 8a
7 = 5u + 12.5a
Simplifying these equations, we get:
4u + 8a = 6 ----(1)
5u + 12.5a = 7 ----(2)
Multiplying equation (1) by 2, we get:
8u + 16a = 12 ----(3)
Subtracting equation (3) from equation (2), we can eliminate 'a':
(5u + 12.5a) - (8u + 16a) = 7 - 12
-3u - 3.5a = -5
Rearranging, we have:
3u + 3.5a = 5 ----(4)
Now we have a system of two equations (1 and 4) with two variables (u and a). We can solve this using any method of solving simultaneous equations.
Taking equation (1):
4u + 8a = 6 ----(1)
Multiplying equation (4) by 8 to eliminate 'a':
24u + 28a = 40 ----(5)
Subtracting equation (1) from equation (5):
(24u + 28a) - (4u + 8a) = 40 - 6
20u + 20a = 34
Dividing by 20:
u + a = 1.7 ----(6)
Subtracting equation (6) from equation (4):
3u + 3.5a - (u + a) = 5 - 1.7
2u + 2.5a = 3.3
Dividing by 2:
u + 1.25a = 1.65
Subtracting equation (6) from this, we can eliminate 'u':
u + 1.25a - (u + a) = 1.65 - 1.7
0.25a = -0.05
Dividing by 0.25:
a = -0.05 / 0.25
a = -0.2 m/s^2
Step 2: Find the initial velocity (u)
Substituting the value of acceleration (a) into equation (6):
u + (-0.2) = 1.7
u = 1.7 + 0.2
u = 1.9 m/s
Therefore, the acceleration of the particle is -0.2 m/s^2 and the initial velocity is 1.9 m/s.