A particle is travelling with velocity (4i+5j)m/s. It undergoes an acceleration of magnitude 2.5m/s^2 in a direction given by the vector (3i-4j). Find the velocity and displacement of the particle from its initial position after 4s.
In the direction of x
u = 4i, a= 2.5(3/5)
At t =4
v = u+at
v= 4 + 2.5 (3/5)4=4+6=10
In the direction of y
u = 5j, a= 2.5(-4/5)
At t =4
v = u+at
v= 5 + 2.5 (-4/5)4=5-8=-3
Hence
V = 10i-3j
Now we have initial and final velocities
r = 1/2(u +v)t
r = 1/2(4i+5j+10i-3j)4
r = 2(14i+2j)
r = 18i+4j
To solve this problem, we can use the equations of motion. Let's start by finding the final velocity of the particle after 4 seconds.
The particle's initial velocity is given as (4i + 5j) m/s.
The acceleration is given as 2.5 m/s^2 in the direction of the vector (3i - 4j).
Acceleration can be written as the derivative of velocity with respect to time:
a = dv/dt
Since acceleration is constant, we can integrate its expression with respect to time to find the change in velocity:
∫ dv = ∫ a dt
Integrating, we get:
Δv = a * t + C
where C is the constant of integration. Since the particle starts with an initial velocity, C will be the initial velocity of the particle.
Plugging in the values:
Δv = (2.5 m/s^2) * (4 s) + (4i + 5j) m/s
= 10i - 10j + 4i + 5j
= 14i - 5j m/s
Hence, the final velocity of the particle after 4 seconds is (14i - 5j) m/s.
Now let's find the displacement of the particle from its initial position after 4 seconds.
Displacement can be obtained by integrating the velocity with respect to time:
∫ ds = ∫ v dt
Again, integrating, we get:
Δs = ∫ v dt
where Δs is the displacement vector.
Integrating each component of the velocity vector separately, we have:
Δx = ∫ (14i - 5j) dt
Δy = ∫ (14i - 5j) dt
The integral of a constant with respect to time is the constant multiplied by the time interval. Hence, the displacement components become:
Δx = (14t)i
Δy = (-5t)j
Now, plugging in t = 4 s, we get:
Δx = (14 * 4)i = 56i
Δy = (-5 * 4)j = -20j
So, the displacement of the particle from its initial position after 4 seconds is 56i - 20j m.