Question

A bicyclist is finishing repair of a flat tire when a friend rides by with a constant speed of 4 m/s. Three seconds later, the bicyclist hops on their bike and accelerates at 2 m/s^2 until he catches the friend. How long does it take the bicyclist to catch the friend from the moment the bicyclist first hops on their bike?

A. 2 seconds
B. 4 seconds
C. 6 seconds
D. 8 seconds

Answer 1

To solve the problem, the distance covered by the friend already in motion before the bicyclist starts pedaling is calculated. Then, a kinematic equation is used to determine when the accelerating bicyclist will cover the same distance as the friend, resulting in a total time of 6 seconds for the bicyclist to catch up.

Step-by-step explanation:

We are tasked with finding out how long it takes a bicyclist to catch up with a friend who is already moving at a constant speed when the bicyclist starts from rest and accelerates. First, let's establish the initial conditions for the friend: The friend is traveling at a constant speed of 4 m/s, and by the time the bicyclist starts, the friend has already covered a distance (since they started 3 seconds earlier).

The distance the friend has traveled when the bicyclist starts is given by:

Distance = Speed × Time = 4 m/s × 3 s = 12 m

Next, let's use the kinematic equation for the bicyclist who starts from rest and accelerates at 2 m/s²:

Distance = Initial Velocity × Time + 0.5 × Acceleration × Time²

Since the initial velocity is 0 m/s for the bicyclist, the equation simplifies to:

Distance = 0.5 × Acceleration × Time² = 0.5 × 2 m/s² × Time²

Now, we need to find when both will have covered the same distance:

12 m + (4 m/s × Time) = 0.5 × 2 m/s² × Time²

12 + 4Time = Time²

Time² - 4Time - 12 = 0

Solving this quadratic equation, we find that Time equals 6 seconds, which is when the bicyclist catches their friend.

Therefore, the correct answer is C. 6 seconds.