a bicyclist is finishing his repair of a flat tire when a friend rides by at 3.5 m/s. two seconds later, the bicyclist hops on his bike and accelerates at 2.4 m/s^2 until he catches his friend? (a) how much time does it take until he catches his friend? (b) how for has he traveled in this time? (c) what is his speed when he catches up?
To solve this problem, we can use the equations of motion. Let's break it down step by step:
(a) How much time does it take until the bicyclist catches his friend?
Step 1: Determine the time it takes for the bicyclist to catch up to his friend after he starts accelerating. Let's call this time t.
Step 2: The friend has already been riding for 2 seconds before the bicyclist starts accelerating. So, the total time it takes for the bicyclist to catch up to his friend would be the time he started accelerating (t) plus the 2 seconds delay.
Step 3: We can use the equation of motion to find the time it takes for the bicyclist to catch up:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
In this case, the initial velocity is 3.5 m/s (the speed of the friend), the acceleration is 2.4 m/s^2 (the acceleration of the bicyclist), and the distance is the same for both the friend and the bicyclist.
So, Distance = (3.5 m/s + 0 m/s) * t + (1/2) * 2.4 m/s^2 * t^2
Simplifying the equation and combining like terms:
0.6 t^2 + 3.5 t = 0
Now, we can solve this quadratic equation to find the value of t.
(b) How far has the bicyclist traveled in this time?
Once we know the time t, we can substitute it back into the equation and find the distance traveled by the bicyclist.
Distance = (3.5 m/s) * t + (1/2) * (2.4 m/s^2) * t^2
(c) What is his speed when he catches up?
To find the speed at which the bicyclist catches up, we can differentiate the equation for distance with respect to time (t). This will give us the instantaneous velocity of the bicyclist at the moment he catches up.
Now that we have the plan to solve the problem, let's proceed with the calculations.