At t = 0, one toy car is set rolling on a straight track with initial position 1

Question

At t = 0, one toy car is set rolling on a straight track with initial position 13.5 cm, initial velocity -2.6 cm/s, and constant acceleration 2.70 cm/s2. At the same moment, another toy car is set rolling on an adjacent track with initial position 8.0 cm, initial velocity 5.40 cm/s, and constant zero acceleration. (a) At what time, if any, do the two cars have equal speeds? (Enter NA if the cars never have equal speeds.) (b) What are their speeds at that time? (Enter NA if the cars never have equal speeds.) (c) At what time(s), if any, do the cars pass each other? (If there is only one time, enter NA in the second blank. If there are two times, enter the smaller time first. If they never pass, enter NA in both blanks.) (d) What are their locations at that time? (If there is only one position, enter NA in the second blank. If there are two positions, enter the smaller position first. If they never pass, enter NA in both blanks.)

Explanation / Answer

here,

for car 1

initial position , x01 = 13.5 cm

initial velocity , u1 = -2.7 m/s

accelration , a1 = 3 cm/s^2

for car 2

initial position , x02 = 9 cm

as car 2 hs zero accelration

therefore , constant velocity , u2 = 5.4 m/s

(a)

let the time t at which both the cars have same speed

using first equation of motion

u2 = u1 + a1 * t

5.4 = -2.7 + 3 * t

t = 2.7 s

the time t at which both the cars have same speed is 2.7 s

(b)

the speed of cars at that time is 5.4 cm/s

(c)

let the position be x at which they pass each other and time be t

for car 1

using seccond equation of motion

(x - x01) = u1t + 0.5 at^2

(x - 13.5) = -2.7 * t + 0.5 * 3 * t^2 ...(1)   

for car 2

( x - x02 ) = u2 * t

( x - 9) = 5.4 * t ...(2)

from equation (1) and (2)

x = 12.39 m , 34.76 m

t = 0.63 s , 4.77 s

the time at which the car pass each other is 0.63 s and 4.77 s

(d)

the location at that time are 12.39 m and 34.76 m at 0.63 s , 4.77 s

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