A car starts from rest and accelerates at a rate of 𝑎𝑐1 = 3.0 𝑚𝑠 in a straight line until it reaches aspeed of 𝑣𝑓 = 25.0 𝑚𝑠 . The car then slows at a constant rate of 𝑎𝑐2 = 2.0 𝑚𝑠 until it stops. (2 sig-figs)a.) How much time elapses from start to stop?b.) How far does the car travel from start to stop?

a.) How much time elapses from start to stop?

First, we need to find out how long it takes for the car to accelerate to its final speed. We can use the formula:

v_f = v_i + a_c1 * t1

where: v_f = final speed = 25.0 m/s v_i = initial speed = 0 (since the car starts from rest) a_c1 = acceleration = 3.0 m/s^2 t1 = time taken to reach final speed

Rearranging the formula to find t1 gives:

t1 = (v_f - v_i) / a_c1 t1 = (25.0 m/s - 0) / 3.0 m/s^2 t1 = 8.33 s

Next, we need to find out how long it takes for the car to decelerate to a stop. We can use the same formula, but with the deceleration rate:

v_f = v_i + a_c2 * t2

where: v_f = final speed = 0 (since the car stops) v_i = initial speed = 25.0 m/s (the speed the car had reached before decelerating) a_c2 = deceleration = -2.0 m/s^2 (deceleration is negative acceleration) t2 = time taken to stop

Rearranging the formula to find t2 gives:

t2 = (v_f - v_i) / a_c2 t2 = (0 - 25.0 m/s) / -2.0 m/s^2 t2 = 12.5 s

The total time elapsed from start to stop is the sum of the time taken to accelerate and the time taken to decelerate:

t_total = t1 + t2 t_total = 8.33 s + 12.5 s t_total = 20.8 s

b.) How far does the car travel from start to stop?

The total distance travelled by the car is the sum of the distance travelled during acceleration and the distance travelled during deceleration.

The distance travelled during acceleration (d1) can be found using the formula:

d1 = v_i * t1 + 0.5 * a_c1 * t1^2

Substituting the known values gives:

d1 = 0 * 8.33 s + 0.5 * 3.0 m/s^2 * (8.33 s)^2 d1 = 104 m

The distance travelled during deceleration (d2) can be found using the formula:

d2 = v_i * t2 + 0.5 * a_c2 * t2^2

Substituting the known values gives:

d2 = 25.0 m/s * 12.5 s + 0.5 * -2.0 m/s^2 * (12.5 s)^2 d2 = 156 m

The total distance travelled from start to stop is the sum of d1 and d2:

d_total = d1 + d2 d_total = 104 m + 156 m d_total = 260 m