7. Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km h–1 in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 m/s2 . If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them?
Solution:
For train A:
Initial velocity, u = 72 km/h = 20 m/s
Time, t = 50 s
Acceleration, a1 = 0 (Since it is moving with a uniform velocity)
From second equation of motion, distance (s1) covered by train A can be obtained as:
s1 = ut + \(\frac 12 a_1t^2\)
= 20 × 50 + 0 = 1000 m
For train B:
Initial velocity, u = 72 km/h = 20 m/s
Acceleration, a = 1 m/s2
Time, t = 50 s From second equation of motion, distance (s11) covered by train A can be obtained as:
s11 = ut + \(\frac 12 at^2\)
= 20 x 50 + \(\frac 12\) x 1 x (50)2 = 2250m
Hence, the original distance between the driver of train A and the guard of train B = 2250 –1000 = 1250 m.
8. On a two-lane road, car A is travelling with a speed of 36 km h–1. Two cars B and C approach car A in opposite directions with a speed of 54 km h–1 each. At a certain instant, when the distance AB is equal to AC, both being 1 km, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident?
Solution:
Velocity of car A, vA = 36 km/h = 10 m/s
Velocity of car B, vB = 54 km/h = 15 m/s
Velocity of car C, vC = 54 km/h = 15 m/s
Relative velocity of car B with respect to car A,
vBA = vB – vA = 15 – 10 = 5 m/s
Relative velocity of car C with respect to car A,
vCA = vC – (– vA) = 15 + 10 = 25 m/s
At a certain instance, both cars B and C are at the same distance from car A i.e.,
s = 1 km = 1000 m
Time taken (t) by car C to cover 1000 m = \(\frac {1000}{25} = 40 s\)
Hence, to avoid an accident, car B must cover the same distance in a maximum of 40 s. From second equation of motion, minimum acceleration (a) produced by car B can be obtained as:
s = ut + \(\frac 12 at^2\)
1000 = 5 x 40 + \(\frac 12\) x a x (40)2
a = \(\frac {1600}{1600}\) =1 m/s2
9. Two towns A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man cycling with a speed of 20 km h–1 in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. What is the period T of the bus service and with what speed (assumed constant) do the buses ply on the road?
Solution:
Let V be the speed of the bus running between towns A and B.
Speed of the cyclist, v = 20 km/h
Relative speed of the bus moving in the direction of the cyclist = V – v = (V – 20) km/h
The bus went past the cyclist every 18 min i.e., \(\frac {18}{60}h\) (when he moves in the direction of the bus).
Distance covered by the bus = (V - 20) \(\frac {18}{60}h\)………….. (i)
Since one bus leaves after every T minutes, the distance travelled by the bus will be equal to:
V x \(\frac {T}{60}\) ……….. (ii)
Both equations (i) and (ii) are equal.
(V - 20) x \(\frac {18}{60} \frac {VT}{60}\)….. (iii)
Relative speed of the bus moving in the opposite direction of the cyclist = (V + 20) km/h
Time taken by the bus to go past the cyclist = 6 min = \( \frac {6}{60}h\)
∴ (V + 20)\( \frac {6}{60}\)=\(\frac {VT}{60}\)….. (iv)
From equations (iii) and (iv), we get
(V + 20) x \( \frac {6}{60}\) = (V - 20) x \(\frac {18}{60}\)
V + 20 = 3V - 60
2V = 80
V = 40km/h
Substituting the value of V in equation (iv), we get
(40 + 20) x \( \frac {6}{60}\) = \(\frac {40T}{60}\)
T = \(\frac {360}{40}\) = 9 min
10. A player throws a ball upwards with an initial speed of 29.4 m s–1 . What is the direction of acceleration during the upward motion of the ball? What are the velocity and acceleration of the ball at the highest point of its motion? Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion. To what height does the ball rise and after how long does the ball return to the player’s hands? (Take g = 9.8 m s–2 and neglect air resistance).
Solution:
The direction of acceleration during the upward motion of the ball is downward.
Velocity = 0, acceleration = 9.8 m/s2
x > 0 for both up and down motions,
v < 0 for up and v > 0 for down motion,
a > 0 throughout the motion 44.1 m, 6 s
Irrespective of the direction of the motion of the ball, acceleration (which is actually acceleration due to gravity) always acts in the downward direction towards the centre of the Earth.
At maximum height, velocity of the ball becomes zero. Acceleration due to gravity at a given place is constant and acts on the ball at all points (including the highest point) with a constant value i.e., 9.8 m/s2 .
During upward motion, the sign of position is positive, sign of velocity is negative, and sign of acceleration is positive. During downward motion, the signs of position, velocity, and acceleration are all positive.
Initial velocity of the ball, u = 29.4 m/s
Final velocity of the ball, v = 0 (At maximum height, the velocity of the ball becomes zero)
Acceleration, a = – g = – 9.8 m/s2
From third equation of motion, height (s) can be calculated as:
v2 - u2 = 2gs
s = \(\frac {v^2-u^2}{2g} = \frac {(o)^2 - (29.4)^2}{2\times (-9.8)}\) = 44.1 m
From first equation of motion, time of ascent (t) is given as:
v = u + at
t = \(\frac {v-u}{a} = \frac {-29.4}{-9.8}\) = 3 s
Time of ascent = Time of descent
Hence, the total time taken by the ball to return to the player’s hands = 3 + 3 = 6 s.