3. At `t = 0`, one toy car is set rolling on a straight track with initial position `15.0 cm`, initial velocity `-3.50 cm//s`, and constant acceleration `2.40 cm//s^2`. At the same moment, another toy car is set rolling on an adjacent track with initial position `10.0 cm`, initial velocity `+5.50 cm//s`, and constant acceleration zero. (a) At what time, if any, do the two cars have equal speeds? (b) What are their speeds at that time? (c) At what time(s), if any, do the cars pass each other? (d) What are their locations at that time? (e) Explain the difference between question (a) and question (c) as clearly as possible.
`x_f = x_i + v_it + 1/2 at^2` quadratic formula: `ax^2 + bx + c = 0` than `x = (-b ± sqrt(b^2 - 4ac))/(2a)`
(a) `3.75\ s` (b) `5.50\ cm//s` (c) `0.604\ s`,`6.90\ s` (d)`13.3\ cm`,`47.9\ cm`
Given: 1st car `x_(i1) = 15cm` `v_(i1) = -3.5 cm//s` `a_1 =2.4 cm//s^2` 2nd car `x_(i2) = 10cm` `v_(i2) = 5.5 cm//s` `a_2 =0` (a) `v_1 = v_(i1) + a_1t` `=> t = (v_1 - v_(i1))/a_1` `v_(i2) = 5.5 cm//s` and keep same speed over time. `v_1 = v_2 = 5.5 cm//s` We get: `t = (5.5 cm//s - (-3.5 cm//s))/(2.4 cm//s^2) = color(red)(3.75s)` (b) When `t=3.75s` `v_1 = -3.5 cm//s + 2.4 cm//s^2 times 3.75 s = color(red)(5.5 cm//s^2)` (c) 1st car `x_1 = x_(i1) + v_(i1)t + 1/2a_1t^2 = 15 - 3.5t -1.2t^2` 2nd car `x_2 = x_(i2) + v_(i12)t = 10 + 5.5t` The passing point is where `x_1 = x_2`. We get: `15 - 3.5t -1.2t^2=10 + 5.5t` `=>1.2t^2 -9t +5 =0 ` Solving for `t` by using the quadratic formula: `t =(9 ± sqrt(9^2 - 4times1.2times5))/(2times1.2)` `color(red)(t=6.9s)` or `color(red)(t=0.604s)` (d) At `6.9s`: `x_1 = x_2 = 10.0 cm + 5.5m//s times 6.9s =color(red)(47.9 cm)` At `6.9s`: `x_1 = x_2 = 10.0 cm + 5.5m//s times 0.604s =color(red)(13.3cm)` (e) At first, `car_1` start form `15.0 cm` moving backward with a forward acceleration. `car_2` start form `10.0cm` moving forward with a constant speed. And 2 cars meet at `x = 13.3cm`. Then, `car_1` stop and change direction with same acceleration. Finally, `car_1` catch up the `car_2` at `x = 47.9m` with a higher speed.