Problem 4- Speedy car, driving at 30 m/s. The driver suddenly observes a slow-moving van 155 m ahead traveling at 5 m/s in the same direction. The driver of the car immediately applies the brakes causing a constant acceleration of 2 m/s2 in a direction opposite to the cars velocity.
a) After the driver of the car applies the brakes, at what time the car will collide with the van?
b) Take x= 0 at the location of the car when the brakes applied, where does the collide with the van?
if the collide, they have traveled the same distance in the same time.
distance van=155+5t
distancecar=30t-1/2 *2*t^2
set them equal, solve for time t.
then using t, solve for car distance.
t = 10.5 s
To solve both parts of the problem, we can use the equations of motion and kinematic equations.
a) To find the time of collision, we need to determine when the distance traveled by the car is equal to the distance between the car and the van.
First, let's calculate the initial relative velocity between the car and the van:
Relative velocity = (velocity of the car) - (velocity of the van)
= 30 m/s - 5 m/s
= 25 m/s
Now, let's calculate the time it takes for the car to cover the distance between the car and the van, which is 155 meters:
Using the equation: distance = initial velocity * time + (1/2) * acceleration * time^2
155 = 25t + (1/2)(-2)t^2
Rearranging the equation:
-2t^2 + 25t - 155 = 0
Solving this quadratic equation will give us the time (t) at which the car will collide with the van.
b) To find the location of the collision, we need to calculate the distance covered by the car from the point when the brakes are applied until the collision occurs.
Using the equation: distance = initial velocity * time + (1/2) * acceleration * time^2
Let's assume the time of collision calculated in part a) is represented by T.
Using this value, calculate the distance covered by the car until the collision:
Distance = 30T + (1/2)(-2)T^2
Now, let's substitute the value of T into the equation to get the location of the collision.