A particle is subject to a time-dependent force: F(t)=5t√. If the particle's mass is 7 kg, what will its velocity be at t=6.7 seconds, assuming its initial velocity at t=0 is equal to 0.
That's F(t)=5sqrt(t)
F(t) = 5√t
F=ma, so
a(t) = 5/7 √t
v(t) = 5/7 * (2/3) t^(3/2) + c
v(0) = 0, so c=0 and
v(t) = 10/21 t^(3/2)
so,
v(6.7) = 10/21 * 6.7^(3/2) = 8.26 m/s
Thanks Steve
To find the velocity of the particle at t=6.7 seconds, we need to integrate the force function F(t) with respect to time and then divide by the particle's mass.
Here's how you can find the velocity:
1. Start by finding the antiderivative of the force function F(t) with respect to time. Since the force function is 5t√, integrate it to get the antiderivative:
∫F(t) dt = ∫(5t√) dt
To integrate t√, use the power rule of integration, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1):
∫F(t) dt = 5∫t√ dt
= 5(√(t^(2+1))/(2+1))
= 5(√(t^3))/3
2. Now that we have the antiderivative of F(t), let's call it V(t) (velocity function), since velocity is the integral of force:
V(t) = 5(√(t^3))/3
3. To find the velocity at t=6.7 seconds, substitute 6.7 into the velocity function V(t):
V(6.7) = 5(√(6.7^3))/3
4. Finally, calculate the velocity by evaluating the expression:
V(6.7) ≈ 9.12 m/s
So, the velocity of the particle at t=6.7 seconds is approximately 9.12 m/s.