A car accelerates at 2.25 m/s^2 along a straight road. It passes two marks that are 27.8 m apart at times t=3.80 s and t=5.25 s. What was the car's velocity at t=0?
Well, to calculate the car's velocity at t=0, we can use the equation v = u + at. Here, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Given that the acceleration is 2.25 m/s^2, and the time is 3.80 s, we can find the final velocity at t=3.80 s.
v = u + at
v = u + (2.25)(3.80)
v = u + 8.55
Now, we also know that the car passes two marks that are 27.8 m apart. Let's assume the car started at x=0 and reached x=27.8 m between t=3.80 s and t=5.25 s.
We can use the equation v^2 = u^2 + 2as to find the final velocity at t=5.25 s.
v^2 = u^2 + 2as
v^2 = u^2 + 2(2.25)(27.8)
v^2 = u^2 + 112.2
v = sqrt(u^2 + 112.2)
Since we want to find the velocity at t=0, we know that the car hasn't moved yet. Therefore, the distance traveled is 0.
Using the equation v = u + at, we have:
0 = u + (2.25)(0)
0 = u + 0
u = 0
So, the car's initial velocity at t=0 is 0 m/s. It hasn't started moving yet, just like me trying to come up with a good joke.
To find the car's velocity at time t=0, we can use the kinematic equation for velocity:
v = u + at
where:
v = final velocity
u = initial velocity (what we are trying to find)
a = acceleration
t = time
Given that the car accelerates at 2.25 m/s^2, we can substitute the values into the equation:
v = u + at
Since we know the velocity at two different times, t=3.80 s and t=5.25 s, we can use these values to find the equation for velocity at those times.
At t=3.80 s:
v_1 = u + a * t_1
At t=5.25 s:
v_2 = u + a * t_2
We can rearrange both equations to isolate the initial velocity:
v_1 - a * t_1 = u
v_2 - a * t_2 = u
Now we can substitute the given values into the equations:
v_1 = velocity at t=3.80 s = ?
a = acceleration = 2.25 m/s^2
t_1 = time at t=3.80 s = 3.80 s
v_2 = velocity at t=5.25 s = ?
t_2 = time at t=5.25 s = 5.25 s
Now we have two equations:
v_1 - 2.25 * 3.80 = u
v_2 - 2.25 * 5.25 = u
We can solve these equations to find the values of v_1 and v_2. Once we have those values, we can substitute them back into the equation to find the initial velocity at t=0:
u = v_1 - 2.25 * 3.80
Now we can calculate the value of u using this equation.