A truck is stopped at a stoplight. When the light turns green, it accelerates at 2.9 m/s2. At the same instant, a car passes the truck going 12 m/s. Where and when does the truck catch up with the car?
The distance S travelled by an object moving at an initial speed v m/s and accelerating at the rate of a m/s² over a period of t sec. is given by
S(t) = ut+(1/2)at²
For the truck:
u=12 m/s
a=0
For the car,
u=0,
a=2.9 m/s²
Equate the distances travelled by the car and the truck, then solve for t (when).
Substitute t into one of the equations to get S (where).
To determine where and when the truck catches up with the car, we can use the equations of motion. Let's break down the problem step by step:
Step 1: Determine the conditions when the truck catches up with the car.
- In this case, the truck catches up with the car when their positions are the same.
Step 2: Write down the equations of motion for both the truck and the car.
- Since the truck is initially at rest, we can use the equation of motion: s = ut + 0.5at^2, where 's' is the distance covered, 'u' is the initial velocity (0 m/s for the truck), 'a' is the acceleration (2.9 m/s^2 for the truck), and 't' is the time.
- For the car, since it is already in motion, we can use the equation: s = ut, where 's' is the distance covered, 'u' is the velocity of the car (12 m/s), and 't' is the time.
Step 3: Set up an equation to find the time when the truck catches up with the car.
- Let's assume 't' is the time it takes for the truck to catch up with the car.
- The distance covered by the truck would be equal to the distance covered by the car, so we can set up an equation: 0 + 0.5(2.9)t^2 = 12t.
Step 4: Solve the equation to find 't'.
- Simplifying the equation: 1.45t^2 = 12t.
- Rearranging the equation: 1.45t^2 - 12t = 0.
- Factoring the equation: t(1.45t - 12) = 0.
- So we have two possible solutions: t = 0 (which is not possible) or t = 12 / 1.45 = 8.28 seconds.
Step 5: Calculate the distance when the truck catches up with the car.
- Using the equation s = ut, we can calculate the distance: s = 12 m/s * 8.28 s = 99.36 meters.
Therefore, the truck catches up with the car after approximately 8.28 seconds at a distance of 99.36 meters.