The locomotive starts off and moves with an acceleration of 1 m / s2 for 10 s, then gradually slows down to zero

The locomotive starts off and moves with an acceleration of 1 m / s2 for 10 s, then gradually slows down to zero for 20 s. What is the average speed of a locomotive along the way?

V0 = 0 m / s.

a1 = 1 m / s2.

t1 = 10 s.

t2 = 20 s.

V2 = 0 m / s.

Vav -?

To find the average speed of movement Vav, it is necessary to divide the entire path traveled by the train S by the time of its movement t along the entire path: Vav = S / t.

S = S1 + S2, where S1 is the acceleration path, S2 is the braking path.

The travel time of the entire path is expressed by the sum: t = t1 + t2, where t1 is the acceleration time, t2 is the deceleration time.

S1 = V0 * t1 + a1 * t1 ^ 2/2.

Since the locomotive starts moving from a state of rest V0 = 0 m / s, the formula will take the form: S1 = a1 * t1 ^ 2/2.

S1 = 1 m / s2 * (10 s) ^ 2/2 = 50 m.

Let’s find the speed at the end of acceleration: V1 = a1 * t1.

V1 = 1 m / s2 * 10 s = 10 m / s.

a ^ 2 = (V1 – V2) / t ^ 2.

a ^ 2 = (10 m / s – 0 m / s) / 20 s = 0.5 m / s2.

S2 = V1 * t2 – a ^ 2 * t2 ^ 2/2.

S2 = 10 m / s * 20 s – 0.5 m / s2 * (20 s) ^ 2/2 = 100 m.

Vav = (50 m + 100 m) / (10 s + 20 s) = 5 m / s.

Answer: the average speed of a locomotive is Vav = 5 m / s.