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 for uniformly accelerated motion. Let's break down the problem step by step:
(a) How much time does it take until he catches his friend?
Step 1: Determine the initial velocity of the bicyclist.
Since the bicyclist was repairing his flat tire, we can assume that his initial velocity was zero.
Step 2: Calculate the time it takes for the bicyclist to catch his friend.
Using the equation of motion: v = u + at, where:
- v is the final velocity (3.5 m/s),
- u is the initial velocity (0 m/s),
- a is the acceleration (2.4 m/s²), and
- t is the time taken,
we can rearrange the equation to solve for t:
t = (v - u) / a
t = (3.5 m/s - 0 m/s) / 2.4 m/s²
Calculating the value, t = 1.46 seconds (rounded to two decimal places).
Therefore, it takes approximately 1.46 seconds for the bicyclist to catch his friend.
(b) How far has he traveled in this time?
To find the distance traveled, we can use the equation of motion: s = ut + 0.5at², where:
- s is the distance traveled,
- u is the initial velocity (0 m/s),
- t is the time taken (1.46 seconds), and
- a is the acceleration (2.4 m/s²).
Substituting the given values into the equation:
s = (0 m/s) * (1.46 seconds) + 0.5 * (2.4 m/s²) * (1.46 seconds)²
After evaluating the equation, we find that the distance traveled is approximately 2.07 meters (rounded to two decimal places).
Therefore, the bicyclist has traveled approximately 2.07 meters in this time.
(c) What is his speed when he catches up?
To calculate the speed of the bicyclist when he catches his friend, we can use the equation of motion: v = u + at, where:
- v is the final velocity (unknown),
- u is the initial velocity (0 m/s),
- a is the acceleration (2.4 m/s²), and
- t is the time taken (1.46 seconds).
Substituting the given values into the equation:
v = (0 m/s) + (2.4 m/s²) * (1.46 seconds)
After calculating, the speed of the bicyclist when he catches up is approximately 3.50 m/s (rounded to two decimal places).
Therefore, the speed of the bicyclist when he catches his friend is approximately 3.50 m/s.