A bullet is fired with a speed of 1000 m/sec in order to hit a target 100 m away. If g = 10 m/s², the gun should be aimed:
(1) Directly towards the target
(2) 5 cm above the target
(3) 10 cm above the target
(4) 15 cm above the target
A body is projected up with a speed 'u' and the time taken by it is T to reach the maximum height H. Pick out the correct statement:
(1) It reaches H/2 in T/2 sec
(2) It acquires velocity u/2 in T/2 sec
(3) Its velocity is u/2 at H/2
(4) Same velocity at 2T
Answer (2)
To answer the first question about aiming the gun:
When a bullet is fired horizontally, it is affected by gravity as it travels to the target. Therefore, the bullet follows a parabolic trajectory, and gravity will cause it to drop by some distance as it travels to the 100 m target.
To find out how high above the target you should aim, you need to calculate the drop due to gravity, which is given by the formula for vertical displacement:
d = 2 1 g t 2
Where:
g = 10 m/s 2 , the acceleration due to gravity.
t is the time it takes for the bullet to reach the target.
The time t can be found using the horizontal motion:
t = v d
Where:
d = 100 m , the distance to the target.
v = 1000 m/s , the speed of the bullet.
Substituting the values:
t = 1000 100 = 0.1 s
Now, calculate the drop d due to gravity:
d = 2 1 ⋅ 10 ⋅ ( 0.1 ) 2 = 0.05 m = 5 cm
Considering this, the gun should be aimed 5 cm above the target. Therefore, the correct answer is (2) 5 cm above the target .
For the second question about the body's motion:
When a body is projected upwards, it moves under the influence of gravity, and its velocity decreases over time until reaching the maximum height.
(2) It acquires velocity 2 u in 2 T sec.
At the maximum height H , the velocity becomes zero. The velocity decreases linearly from u to zero as it ascends.
At halfway in time ( 2 T ), it will have reduced the velocity by half, thus the velocity is 2 u . Therefore, option (2) is correct.
The statements "It reaches H/2 in T/2 sec" , "Its velocity is u/2 at H/2" , and "Same velocity at 2T" are not correct under the constant uniform deceleration due to gravity.
To aim the gun effectively, it should be aimed 5 cm above the target, corresponding to option (2). For the projectile motion question, the body acquires a velocity of u/2 in T/2 seconds, which is option (2).
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