An automobile and a truck start from rest at the same instant, with the automobile initially at some distance behind the truck. The truck has a constant acceleration of 2.15m/s2 , and the automobile an acceleration of 3.47m/s2 . The automobile overtakes the truck after the truck has moved a distance 42.0m .

A) How much time does it take the automobile to overtake the truck?

B) How far was the automobile behind the truck initially?

C) What is the speed of the truck when they are abreast?

D) What is the speed of the automobile when they are abreast?

since s = 1/2 at^2, the truck moved for t seconds, where

42 = 1/2 * 2.15 t^2
t = 6.25 seconds

The car moved
1/2 * 3.47 * 6.25^2 = 67.77 meters
so, it was 25.77 m behind the truck

v = at, so you can figure out the speeds, I expect.

To solve this problem, we can use a set of kinematic equations that relate the initial velocity, final velocity, acceleration, time, and displacement. Here's how we can find the answers to each part of the problem:

A) How much time does it take the automobile to overtake the truck?

Let's assume that the time it takes for the automobile to overtake the truck is t.

For the truck:
Initial velocity, u1 = 0 m/s
Acceleration, a1 = 2.15 m/s^2
Displacement, s = 42.0 m

Using the kinematic equation s = u1t + (1/2)a1t^2, we can solve for t:
42.0 = 0*t + (1/2)*2.15*t^2
42.0 = 1.075t^2
t^2 = 42.0/1.075
t ≈ 8.61 seconds

Therefore, it takes approximately 8.61 seconds for the automobile to overtake the truck.

B) How far was the automobile behind the truck initially?

Since both the automobile and the truck start from rest at the same instant, the initial velocity of the automobile, u2 = 0 m/s.

Using the kinematic equation s = u2t + (1/2)a2t^2, where a2 = 3.47 m/s^2 and t is the time calculated in part A, we can solve for s:
s = 0*t + (1/2)*3.47*t^2
s = 0 + 1.735*t^2
s ≈ 1.735*(8.61)^2
s ≈ 1.735*(74.0521)
s ≈ 128.1 meters

Therefore, the automobile was approximately 128.1 meters behind the truck initially.

C) What is the speed of the truck when they are abreast?

We know that when the automobile overtakes the truck, their displacements are equal.

Let's assume that when they are abreast, the distance covered is d.

For the truck:
Initial velocity, u1 = 0 m/s
Acceleration, a1 = 2.15 m/s^2
Displacement, s1 = d

Using the kinematic equation s = u1t + (1/2)a1t^2, we can solve for t:
d = 0*t + (1/2)*2.15*t^2
d = 1.075t^2

For the automobile:
Initial velocity, u2 = 0 m/s
Acceleration, a2 = 3.47 m/s^2
Displacement, s2 = d

Using the kinematic equation s = u2t + (1/2)a2t^2, we can solve for t:
d = 0*t + (1/2)*3.47*t^2
d = 1.735t^2

Since both expressions equal d, we can equate them and solve for t:
1.075t^2 = 1.735t^2
0.66t^2 = 0
t = 0 (we disregard this solution as it represents the initial rest condition)

Therefore, the time it takes for the truck to have the same displacement as the automobile is 0 seconds.

To find the truck's speed, we can substitute this time into the equation v = u + at, where v is the final velocity of the truck and a1 is its acceleration:
v1 = 0 + 2.15 * 0
v1 = 0 m/s

Therefore, when the automobile overtakes the truck, the truck's speed is 0 m/s.

D) What is the speed of the automobile when they are abreast?

Since they are abreast, we can use the same equation v = u + at to find the automobile's speed. The final velocity, v2, of the automobile is its speed when they are abreast, and u2 is its initial velocity, which is 0 m/s. The acceleration, a2, is given as 3.47 m/s^2.

Using v2 = u2 + a2t, where t = 8.61 seconds (as calculated in part A), we can solve for v2:
v2 = 0 + 3.47 * 8.61
v2 ≈ 29.84 m/s

Therefore, when the automobile overtakes the truck, its speed is approximately 29.84 m/s.

Let's solve the problem step by step:

Step 1: Find the time it takes for the automobile to overtake the truck.

To find the time (t) it takes for the automobile to overtake the truck, we can use the following equation of motion:

s = ut + (1/2)at^2

Where:
s = distance traveled
u = initial velocity
a = acceleration
t = time

Since the automobile starts from rest (u = 0), and we know the acceleration of the automobile (a = 3.47 m/s^2), we can rearrange the equation to solve for time:

s = (1/2)at^2
t^2 = (2s) / a
t = √((2s) / a)

Plugging in the values for distance (s = 42.0 m) and acceleration (a = 3.47 m/s^2):

t = √((2 * 42.0) / 3.47)
t ≈ √(24.208)
t ≈ 4.92 seconds

Therefore, it takes approximately 4.92 seconds for the automobile to overtake the truck.

Step 2: Find the initial distance between the automobile and the truck.

To find the initial distance between the automobile and the truck, we use the equation:

s = ut + (1/2)at^2

Since the truck starts from rest (u = 0), and we know the distance the truck traveled before being overtaken (s = 42.0 m), we can rearrange the equation to solve for the initial distance:

s = (1/2)at^2
2s = at^2
2s / t^2 = a
2(42.0) / (4.92)^2 = a

Calculating the value:

a ≈ 3.45 m/s^2

Therefore, the initial distance between the automobile and the truck is approximately 3.45 m.

Step 3: Find the speed of the truck and automobile when they are abreast.

When the automobile overtakes the truck, they have traveled the same distance. Let's denote this common distance as x.

Since the truck has a constant acceleration (a = 2.15 m/s^2) and starts from rest (u = 0), we can use the following equation of motion to find the speed (v) of the truck:

v^2 = u^2 + 2as

Since the truck starts from rest (u = 0), the equation simplifies to:

v^2 = 2as
v = √(2as)

Plugging in the values for acceleration (a = 2.15 m/s^2) and distance (s = x):

v = √(2 * 2.15 * x)
v = √(4.3x)

Similarly, for the automobile, we have:

v = √(2 * 3.47 * x)
v = √(6.94x)

Since both vehicles are traveling at the same speed (v) when they are abreast, we can set the two equations equal to each other and solve for x:

√(4.3x) = √(6.94x)

Square both sides:

4.3x = 6.94x

Rearrange the equation:

6.94x - 4.3x = 0
2.64x = 0
x = 0

Therefore, when the automobile overtakes the truck (x = 0), their speed is 0 m/s.

Therefore, the speed of the truck and automobile when they are abreast is 0 m/s.