If a body is projected vertically up it's velocity decreases to half of its initial velocity at a height 'h' above the ground The maximum height reached by it is
Ans:4h/3
v = Vi - g t
h = Vi t - .5 g t^2
at height h, v = Vi/2
Vi/2 = Vi - g t
g t = Vi/2
t = Vi/(2 g) at height h
h = Vi^2/(2g) - .5g (Vi^2/4g^2)
h = (1/2)Vi^2/g - (1/8)Vi^2/g
h = (3/8) Vi^2/g
at max height H, v = 0
0 = Vi - g T where T is time at max h
T = Vi/g
H = Vi T -.5 g T^2
H = Vi^2/g -.5 g (Vi^2/g^2)
H = .5 [Vi^2/g] = .5 [8h/3) = (8/6)h
= 4h/3
To find the maximum height reached by the body, we can use the equations of motion for vertical projection. Let's break down the problem and explain the steps to find the answer.
Step 1: Understand the given information
We are given that when a body is projected vertically upwards, its velocity decreases to half of its initial velocity at a height 'h' above the ground. This means that at height 'h', the final velocity (vf) becomes half of the initial velocity (vi).
Step 2: Identify the relevant equations
To solve this problem, we need to use the equations of motion for vertical projection. The equation that relates the final velocity, initial velocity, acceleration due to gravity, and displacement is:
vf^2 = vi^2 + 2gh
where:
vf = final velocity
vi = initial velocity
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height
We can also use the equation for finding the final velocity in terms of initial velocity, acceleration due to gravity, and displacement:
vf = vi + gt
Step 3: Set up the equations using the given information
Given that the final velocity decreases to half of the initial velocity at height 'h', we can write the equation as:
vf = (1/2)vi
Using the second equation of motion, we can write:
(1/2)vi = vi + gt
Step 4: Solve the equations
Let's solve the equation to find the relation between 'h' and 'vi':
(1/2)vi = vi + gt
Multiply both sides of the equation by 2 to eliminate the fractions:
vi = 2vi + 2gt
Rearrange the equation:
vi = -2gt
Now, using the equation vf = vi + gt, we can find the final velocity at height 'h':
vf = vi + gt
Substituting the value of vi from the previous equation, we get:
vf = -2gt + gt
For vf to be equal to half of vi, we set:
-2gt + gt = (1/2)vi
Simplify the equation:
-gt = (1/2)vi
Divide both sides of the equation by 'g':
-t = (1/2)vi/g
Since time 't' represents the time taken to reach height 'h', we can express it in terms of 'h' and the initial velocity 'vi'. Using the equation for displacement, h = vi(t) - (1/2)gt^2, we can rewrite it as:
h = vit - (1/2)gt^2
Substituting the value of 't' from the equation above, we get:
h = vi((1/2)vi/g) - (1/2)g((1/2)vi/g)^2
Simplifying the equation:
h = (vi^2)/(2g)
Therefore, the maximum height reached by the body is (vi^2)/(2g).
Step 5: Calculate the answer
Given the answer is 4h/3, we can equate it to (vi^2)/(2g) and solve for h:
4h/3 = (vi^2)/(2g)
Multiply both sides of the equation by (3/4) to solve for h:
h = (3/4)(vi^2)/(2g)
Which simplifies to:
h = (3/8)(vi^2)/g
Comparing this equation with the given answer, we can conclude that:
(3/8)(vi^2)/g = 4h/3
Therefore, the answer 4h/3 is correct.
So, the maximum height reached by the body is 4h/3.