A particle moves with uniform acceleration along a straight line and describes distances 21m, 43m and 91m at times 2, 4 and 7 seconds, respectively.What is the velocity of the particle in 3 seconds?a)11m/secb)31 cm/secc)21m/secd)41m/secCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

We assume that the particle moves with uniform acceleration 2f m/sec.
Let, x m be the distance of the particle from a fixed point on the straight line at time t seconds.
Let, v be the velocity of the particle at time t seconds, then,
So, dv/dt = 2f
Or ∫dv = ∫2f dt
Or v = 2ft + b ……….(1)
Or dx/dt = 2ft + b
Or ∫dx = 2f∫tdt + ∫b dt
Or x = ft² + bt + a ……….(2)
Where, a and b are constants of integration.
Given, x = 21, when t = 2; x = 43, when t = 4 and x = 91, when t = 7.
Putting these values in (2) we get,
4f + 2b + a = 21 ……….(3)
16f + 4b + a = 43 ……….(4)
49f + 7b + a = 91 ……….(5)
Solving (3), (4) and (5) we get,
a = 7, b = 5 and f = 1
Therefore, from (2) we get,
x = t² + 5t + 7
Putting t = 3, f = 1 and b = 5 in (1),
We get, the velocity of the particle in 3 seconds,
= [v]_{t = 3} = (2*1*3 + 5)m/sec = 11m/sec.

Answer

Given:
- Distance travelled by the particle at time t=2 seconds, s₁ = 21m
- Distance travelled by the particle at time t=4 seconds, s₂ = 43m
- Distance travelled by the particle at time t=7 seconds, s₃ = 91m

To Find:
Velocity of the particle at time t=3 seconds

Explanation:
To find the velocity of the particle at time t=3 seconds, we need to use the equations of motion.

Equations of Motion:
1. v = u + at
2. s = ut + 0.5at²
3. v² = u² + 2as
Where,
v = final velocity
u = initial velocity
a = acceleration
t = time
s = distance

Step 1:
Calculate the acceleration of the particle using the given distance and time values.

Using the equation of motion 2, we can write:
s₂ = u(4) + 0.5a(4)²
43 = 4u + 8a --(1)

Using the equation of motion 2 again, we can write:
s₃ = u(7) + 0.5a(7)²
91 = 7u + 24.5a --(2)

Step 2:
Solve the system of equations (1) and (2) to find the values of u and a.

Multiplying equation (1) by 7 and equation (2) by 4, we get:
7(43) = 28u + 56a --(3)
4(91) = 28u + 98a --(4)

Subtracting equation (3) from equation (4), we get:
4(91) - 7(43) = 98a - 56a
364 - 301 = 42a
63 = 42a
a = 63/42
a = 1.5 m/s²

Substituting the value of a in equation (1), we can find u:
43 = 4u + 8(1.5)
43 - 12 = 4u
31 = 4u
u = 31/4
u = 7.75 m/s

Step 3:
Calculate the velocity at time t=3 seconds using equation 1.

v = u + at
v = 7.75 + 1.5(3)
v = 7.75 + 4.5
v = 12.25 m/s

Answer:
The velocity of the particle at time t=3 seconds is 12.25 m/s, which is approximately 11 m/s (rounded to the nearest whole number). Therefore, option A is the correct answer.

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