Graphs
Ques 28: Displacementtime graph of a particle moving in a straight line is as shown in figure.
(a) Find the sign of velocity in regions oa, ab, be and cd.
(b) Find the sign of acceleration in the above region.
Ans: (a) positive, positive, positive, negative (b) positive, zero, negative, negative
Sol: OA : slope is + ive and increasing.
∴ velocity is + ive and acceleration is + ive.
AB : slope is + ive and constant
∴ velocity is +ive and acceleration is zero.
BC : sope is + ive and decreases.
∴ velocity is + ive and increasing.
CD : slope is –ive and increasing
∴ velocity is  ive and acceleration is  ive.
Ques 29: Let us call a motion as:
M_{1} → if velocity and acceleration both are positive.
M_{2} → if velocity is positive but acceleration is negative.
M_{3 }→ if velocity and acceleration both are negative.
M_{4} → if velocity is negative but acceleration is positive.
(a) State, in which of the above four motions, magnitude of velocity is increasing and in which it is decreasing.
(b) vt graph of a particle moving in a straight line is as shown in figure. The whole graph is made up of four straight lines P, Q, R and S. These four straight lines indicate four type of motions (M_{1}.. , M_{4}) discussed above. State, which straight line corresponds to which type of motion.
Ans: (a) In M_{1} and M_{3} magnitude is increasing, in M_{2} and M_{4} magnitude is decreasing
(b) P → M_{1}; Q → M_{2}; R → M_{3}; S → M_{4}
Sol:
In M_{1} and M_{3} : 0° < θ < 90°
∴ slope is + ive i.e., acceleration is + ive.
In M_{2} and M_{4} : 90° < θ < 180°
∴ slope is  ive i.e., acceleration is  ive.
(a) M_{1} : Magnitude of velocity is increasing.
M_{2} : Magnitude of velocity is decreasing.
M_{3} : Magnitude of velocity is increasing.
M_{4} : Magnitude of velocity is decreasing.
M_{1} and M_{3}.
(b) P → M_{1}
Q → M_{2}
R → M_{3}
S → M_{4}
Ques 30: Velocitytime graph of a particle moving in a straight line is shown in figure. At time t = 0, s =  10 m. Plot corresponding at and st graphs.
Ans:
Sol: Case I
v = mt  v_{0} (stline)
∴ ∫ ds = ∫ (mt  v_{0}) dt
As for 0° < q < 90°
tan θ is + ive, a is + ive.
Case II
v = k (constant) (st line parallel to timeaxis)
⇒ ∫ ds = ∫ k dt
s = kt + s_{0}
Further, a = dv/dt = 0
Case III.
Timedisplacement graph  
Time  Area under the graph  Initial Displacement  Net displacement 
at 0 s  0 m   10 m   10 m 
2 s  10 m   10 m  0 m 
4 s  30 m   10 m  + 20 m 
6 s  40 m   10 m  + 30 m 
8 s  30 m   10 m  + 20 m 
10 s  20 m   10 m  + 10 m 
Timeacceleration graph  
time  slope  acceleration 
0  2 s  5  5 m/s^{2} 
2  4 s  zero  0 m/s^{2} 
4  6 s   5   5 m/s^{2} 
6  8 s   5   5 m/s^{2} 
8  10 s  + 5  + 5 m/s^{2} 
Ques 31: Velocitytime graph of a particle moving in a straight line is shown in figure. At time t = 0, s = 20 m. Plot at and st graphs of the particle.
Ans:
Sol: a = slope of vt graph.
S = area under vt graph.
Corresponding graphs are drawn in the answer sheet.
Ques 32: Velocitytime graph of a particle moving in a straight line is shown in figure. In the time interval from t = 0 to t = 14s, find:
(a) average velocity and
(b) average speed of the particle
Ans: (a) (50/7) ms^{1}
(b) 10 ms^{1}
Sol:
Ques 33: Accelerationtime graph of a particle moving in a straight line is as shown in figure. At time t = 0, velocity of the particle is zero. Find:
(a) average acceleration in a time interval from t = 6s to t = 12s,
(b) velocity of the particle at t = 14s.
Ans: (a) 5 ms^{2}
(b) 90 ms^{1}
Sol: Average acceleration
v_{f}  v_{i} = Area under at graph.
Ques 34: Velocitytime graph of a particle moving in a straight line is shown in figure. At time t = 0, displacement of the particle from mean position is 10 m. Find:
(a) acceleration of particle at t = Is, 3s and 9 s.
(b) position of particle from mean position at t = 10s.
(c) write down st equation for time interval:
(i) 0 < / < 2s,
(ii) 4s < t < 8s
Ans: (a) 5 ms^{2}, zero, 5 ms^{2}
(b) s = 30 m
(c) (i) s = 10 + 2.5 t^{2}
(ii) s = 40 + 10 (t  4)  2.5 (t  4)^{2}
Sol: (a) Acceleration = slope of vt graph.
(b) r_{f}  r_{i} = s = area under vt graph.
(c) Equations are written in answer sheet.
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