the displacement (in meter) of a particle moving in a straight line is given by the equation of motion s=5t^3+4t+2, where t is measured in seconds. Find the velocity of the particle at t=3.
To find the velocity of the particle at a specific time, we need to find the derivative of the displacement equation with respect to time. The derivative of the displacement function will give us the velocity function.
Given that the displacement equation is s = 5t^3 + 4t + 2, we can find the velocity at a specific time t by taking the derivative of this equation.
Let's start by finding the derivative of the displacement equation, s, with respect to time, t.
ds/dt = d/dt (5t^3 + 4t + 2)
The derivative of t^n, where n is a constant, can be found by multiplying the constant by the power of t and reducing the power by 1.
So, the derivative of t^3 is 3t^(3-1) = 3t^2
The derivative of t^1 is 1t^(1-1) = 1
Therefore, the derivative of the displacement equation is:
ds/dt = 15t^2 + 4
Now that we have the velocity function ds/dt, we can substitute the given time, t = 3, to find the velocity of the particle.
Velocity at t = 3:
v = (ds/dt) at t = 3
= 15(3)^2 + 4
= 135 + 4
= 139
Therefore, the velocity of the particle at t = 3 seconds is 139 m/s.