Hello, Anooshka
The problem is asking for total displacement over the period 0-5 seconds. We've already been given the equation for velocity for this time, and, we know that displacement is the integral of velocity. So, to solve this problem we just need to take the integral of velocity from t=0 to t=5.
D(t) = ∫V(t)dt where V(t) = -2t-1
D(t) = ∫(-2t-1)dt which is a simple power rule integration: ∫f(t)ndt = F(t)n+1/(N+1) + C So,
D(t) = ∫(-2t-1)dt = (-2/2)t2 - (1/1)t + C which reduces to D(t) = -t2 - t
Now, we know that the bounds for this displacement are t = 0, and t = 5. So we just need to evaluate this indefinite integral at those two times and subtract the difference to find the amount that the particle traveled.
D(5) - D(0) where D(5) = -(5)2 - (5) = -30 and D(0) = -(0)2 - 0 = 0
So, D(5) - D(0) = -30 - 0 = -30
The particle thus was displaced -30.