An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 17.0 m if her initial speed is 3.80 m/s. What is the free-fall acceleration on the planet?
The horizontal distance traveled is given by d = V^2(sin(2µ))/g where V = the initial velocity, g = the acceleration due to gravity and µ = the initial angle of the velocity to the horizontal.
Since the maximum d will occur when µ = 45º. Therefore, d = 17 = 3.8^2(1)g. Solve for g.
To find the free-fall acceleration on the planet, we can use the horizontal distance and the initial speed of the astronaut during the jump. Let's break down the problem to understand the steps needed to solve it.
The horizontal distance covered during the jump by the astronaut is 17.0 m, and her initial speed is 3.80 m/s. We can use the equation for horizontal motion:
distance = speed * time
Since there are no horizontal forces acting on the astronaut during the jump, her initial speed will remain constant throughout. Therefore, the time of flight will be the same as the time it takes her to cover the horizontal distance.
We can use the equation to find the time of flight:
time = distance / speed
Substituting the given values:
time = 17.0 m / 3.80 m/s
time = 4.47 s (rounded to two decimal places)
Now, we know the time of flight during the jump. During free fall, only the vertical motion is affected by gravity. The equation for vertical motion is:
height = initial_velocity * time + 1/2 * acceleration * time^2
At the maximum height, the final velocity is 0 m/s because the astronaut is momentarily at rest. Therefore, the equation becomes:
0 = initial_velocity * time + 1/2 * acceleration * time^2
We can rearrange the equation to solve for the acceleration:
acceleration = - (2 * initial_velocity * time) / (time^2)
Substituting the given values:
acceleration = - (2 * 3.80 m/s * 4.47 s) / (4.47 s)^2
acceleration = - (8.54 m) / (19.98 s^2)
acceleration = -0.428 m/s^2 (rounded to three decimal places)
Therefore, the free-fall acceleration on the planet is approximately -0.428 m/s^2. The negative sign indicates that it is directed opposite to the positive direction of motion.
To find the free-fall acceleration on the planet, we can use the equation for the range of a projectile motion:
Range = (Initial speed^2 / acceleration due to gravity) * sin(2θ)
Where:
- Range is the maximum horizontal distance covered
- Initial speed is the speed at which the astronaut jumps horizontally
- Acceleration due to gravity is the free-fall acceleration on the planet
- θ is the angle of projection (which is 45 degrees for maximum range)
Given:
- Range = 17.0 m
- Initial speed = 3.80 m/s
- θ = 45 degrees
We can rearrange the equation to solve for the acceleration due to gravity:
Acceleration due to gravity = (Initial speed^2 / (Range * sin(2θ)))
Plugging in the values:
Acceleration due to gravity = (3.80 m/s)^2 / (17.0 m * sin(2 * 45 degrees))
Simplifying:
Acceleration due to gravity = 14.44 m^2/s^2 / (17.0 m * sin(90 degrees))
Since sin(90 degrees) = 1, the equation becomes:
Acceleration due to gravity = 14.44 m^2/s^2 / 17.0 m
Finally, dividing the values:
Acceleration due to gravity ≈ 0.85 m/s^2
Therefore, the free-fall acceleration on the planet is approximately 0.85 m/s^2.