a particle is travelling in a straight line with constant acceleration.it covers 6m in the fourth second and 7m in the fifth second.find its acceleration and initial speed.
To find the acceleration and initial speed of the particle, we can use the equations of motion.
The first equation of motion is:
\[s = ut + \frac{1}{2}at^2\]
where:
- s is the displacement
- u is the initial velocity
- t is the time
- a is the acceleration
Using this equation, we can calculate the displacement of the particle in the fourth and fifth seconds.
In the fourth second:
\[6 = u \times 4 + \frac{1}{2} \times a \times 4^2\]
\[6 = 4u + 8a\] ---(1)
In the fifth second:
\[7 = u \times 5 + \frac{1}{2} \times a \times 5^2\]
\[7 = 5u + \frac{25}{2}a\] ---(2)
We now have a system of two equations (equations 1 and 2) with two unknowns (u and a). We can solve this system of equations to find the values of u and a.
First, let's multiply equation (1) by 5, and equation (2) by 4 to eliminate the u term:
\[5(6) = 5(4u) + 5(8a)\]
\[7(7) = 7(5u) + 7(\frac{25}{2}a)\]
Simplifying, we get:
\[30 = 20u + 40a\] ---(3)
\[49 = 35u + \frac{175}{2}a\] ---(4)
Now, let's subtract equation (3) from equation (4):
\[49 - 30 = 35u - 20u + \frac{175}{2}a - 40a\]
\[19 = 15u + \frac{95}{2}a\]
To simplify this equation, let's multiply it by 2 to eliminate the fraction:
\[38 = 30u + 95a\] ---(5)
Now, we have two equations, (3) and (5), with two unknowns, u and a.
Equation (3): 30 = 20u + 40a
Equation (5): 38 = 30u + 95a
Subtracting equation (3) from equation (5), we get:
\[38 - 30 = 30u - 20u + 95a - 40a\]
\[8 = 10u + 55a\]
Simplifying:
\[8 = 10(u + 5.5a)\]
\[u + 5.5a = \frac{8}{10}\]
\[u + 5.5a = 0.8\] ---(6)
Now, let's substitute the value of (u + 5.5a) from equation (6) into equation (3):
\[30 = 20(0.8) + 40a\]
\[30 = 16 + 40a\]
\[14 = 40a\]
\[a = \frac{14}{40}\]
\[a = 0.35 \, \text{m/s}^2\]
Substituting this value of a back into equation (6) to find u:
\[u + 5.5(0.35) = 0.8\]
\[u + 1.925 = 0.8\]
\[u = 0.8 - 1.925\]
\[u = -1.125 \, \text{m/s}\]
Therefore, the acceleration of the particle is \(0.35 \, \text{m/s}^2\) and the initial velocity is \(-1.125 \, \text{m/s}\).