A particle is projected vertically upward with a velocity of
10 ms-1. The velocity of the particle when it again reaches
the point of projection after 2 seconds is (neglecting air
resistance).
(a) 0
(b) 10 m s-1
(c) 20 m s-1 (d) 5 ms-1
Answers
Answer
Using the scalar method
First, the particle moves up and reaches the highest point. Then the particle moves towards downward direction and finally strikes ground after 5 s.
At the highest point, the speed of the particle will be zero.
Here speed is in upward direction and acceleration is in the downward direction. Hence, the sign of acceleration should be negative.
Using v
2
=u
2
+2as
Hence, 0 = (10)
2
−2×10×H
1
⇒ H
1
= 5m
Using v = u + at from A to B,
0 = 10 - 10 ×t
1
⇒
t
1
= 1s
At highest point, the velocity of the particle will (a) be zero.
The particle starts moving towards downward direction. Here the velocity and acceleration both are in downward direction. Hence, the sign of acceleration should be positive.
The time taken by the particle from B to C will be 5 - 1 = 4 s
Now using s =ut +
2
1
at
2
from B to C
(H
1
+H)=(1/2)×10×4
2
= 80m
5 + H = 80m
⇒ Hence height of the building H = 75m
Using vector method
The particle starts from A and finally reaches at C.
Let us take the origin at A. The upward direction is taken as positive and the downward direction is taken as negative.
The particle moves in a gravitational field where the acceleration due to gravity is always acting in the downward direction whether it is moving upward or downward.
Hence, acceleration vector
a
will always be -10 ms−2, as its magnitude, as well as direction, remain constant always throughout the motion.
Hence, acceleration
a
= -10 ms
−2
Initially at t = 0, the particle is projected in the upward direction. Hence, initial velocity
u
= 10 ms−1. The particle moves from A to B (upward) and then B to C (downward).
The motion of the particle from A to B then again passes point A. The net displacement of the particle upto this instant is zero. Then particle crosses point A and finally reaches to C. We know net displacement is equal to the difference of final position vector and the initial position vector. Hence, net displacement of the particle during motion (t = 5 s) is -H (m).
Using
s
=
u
t+(1/2)
a
t
2
=(10)×5+(1/2)(−10)(5)
2
= 50 - 125 = - 75 (m)
Hence, H = 75m.
Answer:
Using the scalar method
First, the particle moves up and reaches the highest point. Then the particle moves towards downward direction and finally strikes ground after 5 s.
At the highest point, the speed of the particle will be zero.
Here speed is in upward direction and acceleration is in the downward direction. Hence, the sign of acceleration should be negative.
Using v
2
=u
2
+2as
Hence, 0 = (10)
2
−2×10×H
1
⇒ H
1
= 5m
Using v = u + at from A to B,
0 = 10 - 10 ×t
1
⇒
t
1
= 1s
At highest point, the velocity of the particle will (a) be zero.
The particle starts moving towards downward direction. Here the velocity and acceleration both are in downward direction. Hence, the sign of acceleration should be positive.
The time taken by the particle from B to C will be 5 - 1 = 4 s
Now using s =ut +
2
1
at
2
from B to C
(H
1
+H)=(1/2)×10×4
2
= 80m
5 + H = 80m
⇒ Hence height of the building H = 75m