A parachutist jumps from height 100m. He wants to reach at ground with zero velocity. For this purpose he switches on a parachute propeller after falling freely for certain height. Given that the parachute propeller is switched on total acceleration of the man varies with velocity as a = -2v, where v is instantaneous velocity of the man. Find the time after falling freely man should switch on parachute propeller for this purpose.
Can't you make it little easier ? We asked this question so that we can solve this difficult question but this solution is more difficult than the question itself.
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To solve this problem, we need to find the height at which the parachutist should switch on the parachute propeller in order to land on the ground with zero velocity.
Let's break down the problem into steps:
Step 1: Find the time it takes for the parachutist to reach the desired switching height.
To find this time, we can use the equation of motion:
h = 1/2 * g * t^2
where h is the height, g is the acceleration due to gravity (9.8 m/s^2), and t is the time.
Since the parachutist falls freely before switching on the parachute propeller, we can consider the free-fall equation:
h = 1/2 * g * t^2
Substituting the values: h = 100 m and g = 9.8 m/s^2, we can solve for t:
100 = 1/2 * 9.8 * t^2
Simplifying the equation:
t^2 = 100 / (1/2 * 9.8)
t^2 = 20.408
Taking the square root of both sides:
t = sqrt(20.408)
t ≈ 4.52 s
So, it takes approximately 4.52 seconds for the parachutist to fall freely.
Step 2: Find the height at which the parachute propeller should be switched on.
The given acceleration with respect to velocity is:
a = -2v
The total acceleration of the parachutist is the sum of the acceleration due to gravity (g) and the acceleration due to the propeller (a):
a_total = g + a
At the point of switching on the parachute propeller, the total acceleration should be equal to zero, since the parachutist wants to reach the ground with zero velocity. Therefore:
0 = g + a
0 = 9.8 + (-2v)
Solving for v:
2v = 9.8
v = 4.9 m/s
Step 3: Find the height at which the parachute propeller should be switched on.
To find the height at which the parachute propeller should be switched on, we can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s), u is the initial velocity (4.9 m/s), a is the total acceleration, and s is the distance traveled.
Substituting the known values, and rearranging the equation:
0 = (4.9)^2 + 2a * s
Simplifying the equation:
s = -(4.9)^2 / (2a)
Substituting the given value of a = -2v:
s = -(4.9)^2 / (2 * (-2 * 4.9))
s ≈ -6.05 m
Since we are interested in the height from the ground, we take the absolute value of s:
s ≈ 6.05 m
Therefore, the parachutist should switch on the parachute propeller approximately 6.05 meters above the ground to land with zero velocity.
Note: The negative sign in the distance indicates that it is below the starting point.
since dv/dt = -2v
v(t) will be an exponential function, so it will never be zero. You sure you don't want to specify some small acceptable speed at touchdown?
Anyway, consider that after falling for k seconds,
v = -9.8k
and after t more seconds when s=0, you want v close to zero
v = -9.8k e^(-2t)
s = 100 - 9.8kt + 1/2 at^2
= 100 - 9.8kt - 9.8k*e^(-2t)*t^2
See what you can do with that.