On the moon the acceleration due to gravity is 5 ft/sec2. An astronaut jumps into the air with an initial upward velocity of 8 ft/sec.
How high does she/he go?
How long is the astronaut off the ground?
To find the height the astronaut reaches and the time they are off the ground, we can use the equations of motion. Let's break down the problem step by step:
1. Determine the time it takes for the astronaut to reach the highest point:
We can use the kinematic equation: v = u + at, where:
- v is the final velocity (0 ft/sec, when the astronaut reaches the highest point),
- u is the initial velocity (8 ft/sec, upwards),
- a is the acceleration due to gravity on the moon (5 ft/sec^2), and
- t is the time it takes to reach the highest point.
Rearranging the equation, we have 0 = 8 - 5t. Solving for t gives t = 8/5 seconds.
2. Calculate the height the astronaut reaches:
We use the kinematic equation: s = ut + (1/2)at^2, where:
- s is the displacement (height above the ground),
- u is the initial velocity (8 ft/sec, upwards),
- t is the time to reach the highest point (8/5 seconds), and
- a is the acceleration due to gravity on the moon (5 ft/sec^2).
Substituting the values, we have s = (8)(8/5) + (1/2)(5)(8/5)^2.
Simplifying, s = 12.8 + 12.8 = 25.6 feet.
Therefore, the astronaut reaches a height of 25.6 feet.
3. Calculate the total time the astronaut is off the ground:
Since the astronaut comes back down to the ground, we can double the time calculated in step 1 to find the total time off the ground.
t_total = 2 * (8/5) = 16/5 seconds.
Therefore, the astronaut is off the ground for 16/5 seconds or 3.2 seconds.
To find out how high the astronaut goes, we can use the kinematic equation for displacement in freefall:
h = (v0^2) / (2 * g)
Where:
h = height
v0 = initial velocity
g = acceleration due to gravity
Given that the initial upward velocity is 8 ft/sec and the acceleration due to gravity on the moon is 5 ft/sec^2, we can substitute the values into the equation:
h = (8^2) / (2 * 5)
h = 64 / 10
h = 6.4 ft
Therefore, the astronaut reaches a height of 6.4 ft.
To find out how long the astronaut is off the ground, we can use the kinematic equation for time:
t = (v - v0) / g
Where:
t = time
v = final velocity
v0 = initial velocity
g = acceleration due to gravity
Since the final velocity is zero at the highest point, we can substitute the values into the equation:
t = (0 - 8) / (-5)
t = -8 / -5
t = 1.6 seconds
Therefore, the astronaut is off the ground for 1.6 seconds.
h(t) = 8t - 2.5t^2
so, just find the vertex of that parabola.