Question

A woman on a motorcycle moving uniformly at a rate of 20 m/s passes a truck at rest. At the instant the motorcycle passes the truck, the truck begins to accelerate at the rate of 5 m/s . (A) How long does it take the truck to catch up to the motorcycle? (B) How far has the motorcycle traveled? (C) What is the truck's speed at this point?

Answer 1

Answer:

A) It takes the truck 8 s to catch the motorcycle.

B) The motorcycle has traveled 160 m in that time.

C) The velocity of the truck is 40 m/s at that time.

Explanation:

The equations of the position and velocity of an object moving in a straight line are as follows:

x = x0 +v0 · t + 1/2 · a · t²

v = v0 + a · t

Where:

x = position

x0 = initial position

v0 = initial velocity

t = time

a = acceleration

v = velocity at time t

(A) When the the truck catches the motorcycle, both have the same position. Notice that the motorcycle moves at constant speed so that a = 0:

x truck = x motorcycle

x0 +v0 · t + 1/2 · a · t² = x0 + v · t

Placing the origin of the frame of reference at the point where the truck starts, both have an initial position of 0. The initial velocity of the truck is 0. Then:

1/2 · a · t² = v · t

solving for t:

t = 2 v/a

t = 2 · 20 m/s/ 5 m/s²

t = 8 s

It takes the truck 8 s to catch the motorcycle.

(B) Using the equation of the position of the motorcycle, we can calculate the traveled distance in 8 s.

x = v · t

x = 20 m/s · 8 s

x = 160 m

(C) Now, we use the velocity equation at time 8 s.

v = v0 + a · t

v = 0 m/s + 5 m/s² · 8 s

v = 40 m/s