The method by which the Mars exploration rovers, Spirit and Opportunity, landed on the surface of Mars in January of 2004 was quite elaborate. The rovers began their descent through the thin Martian atmosphere on a parachute until they reached an altitude of about 16.7 m. At that point a system of four air bags with six lobes each were inflated, retro rockets
brought the craft to rest, and the rovers detached from their parachutes.
After free-falling to the surface, with an acceleration of 3.72 m/s^2, and bouncing about a dozen times, the rovers deflated their air bags, righted themselves, and went about their explorations. On the first bounce from the surface, the rovers had an initial velocity of 9.92 m/s at an angle of 75.0
degrees above the horizontal. (A) What was the maximum height of a rover between the first and second bounces? (B) How far did a rover travel in the horizontal direction between the first and second bounces?
2.57
To solve this problem, we will use the principles of projectile motion.
(A) To find the maximum height of a rover between the first and second bounces, we need to find the time it takes for the rover to reach the highest point in its trajectory. Let's break down the given information:
Initial velocity (v₀) = 9.92 m/s
Angle (θ) = 75.0 degrees
Acceleration due to gravity (g) = 9.8 m/s² (assuming it is the same on Mars as on Earth)
First, let's resolve the initial velocity into horizontal and vertical components:
Horizontal component: v₀x = v₀ * cos(θ)
Vertical component: v₀y = v₀ * sin(θ)
v₀x = 9.92 m/s * cos(75.0°) = 2.49 m/s
v₀y = 9.92 m/s * sin(75.0°) = 9.46 m/s
Next, let's find the time it takes for the rover to reach its maximum height. We can use the following equation:
v_y = v₀y + a * t
Since the final velocity at the top is 0 (v_y = 0), we have:
0 = 9.46 m/s + (-9.8 m/s²) * t
Solving for t:
t = 9.46 m/s / 9.8 m/s²
t ≈ 0.9649 s
Now, we can find the maximum height using the equation:
Δy = v₀y * t + (1/2) * a * t^2
Δy = 9.46 m/s * 0.9649 s + (1/2) * (-9.8 m/s²) * (0.9649 s)^2
Δy ≈ 4.577 m
Therefore, the maximum height of a rover between the first and second bounces is approximately 4.577 meters.
(B) To find the horizontal distance traveled by the rover between the first and second bounces, we can use the equation:
Δx = v₀x * t
Δx = 2.49 m/s * 0.9649 s
Δx ≈ 2.4033 m
Therefore, the rover traveled approximately 2.4033 meters in the horizontal direction between the first and second bounces.
To find the answer to part (A), we need to determine the maximum height reached by the rover between the first and second bounces.
Let's break down the problem:
1. We know the initial velocity (v0) of the rover, which is 9.92 m/s at an angle of 75.0 degrees above the horizontal. This can be decomposed into horizontal and vertical components using trigonometry.
Horizontal component: v0x = v0 * cos(theta)
Vertical component: v0y = v0 * sin(theta)
2. We need to find the time it took for the rover to reach its maximum height. This can be done using kinematic equations.
The vertical motion equation is:
y = y0 + v0yt - (1/2)gt^2
Where:
y = final vertical position (maximum height)
y0 = initial vertical position (which we can assume to be 0)
v0y = initial vertical velocity
g = acceleration due to gravity on Mars (approximately 3.72 m/s^2)
t = time
We want to find the time (t) at maximum height, so we set v0y = 0 in the equation.
0 = v0y - gt_max_height
t_max_height = v0y / g
3. Once we have the time at maximum height, we can find the maximum height (y_max_height) by substituting it back into the equation.
y_max_height = y0 + v0yt_max_height - (1/2)gt_max_height^2
Now let's calculate the values:
v0x = 9.92 m/s * cos(75.0 degrees)
v0y = 9.92 m/s * sin(75.0 degrees)
t_max_height = v0y / g
y_max_height = (0) + (v0y) * (t_max_height) - (1/2) * g * (t_max_height^2)
To find the answer to part (B), we need to determine the horizontal distance traveled by the rover between the first and second bounces.
Horizontal distance (x) can be calculated using the equation:
x = v0x * t
Where:
v0x = initial horizontal velocity
t = total time of flight from the first to second bounce
Now let's calculate the values:
v0x = 9.92 m/s * cos(75.0 degrees)
t = 2 * t_max_height (since the time of flight is symmetric)
x = v0x * t
By following these steps and calculations, you will be able to find the maximum height (A) and the horizontal distance (B) traveled by the rover between the first and second bounces.
(A) Use the vertical component of the initial bounce velocity (9.92 m/s*sin 75) and the acceleration of gravity near the Martian surface (3.72 m/s^2) to get the time in the air during the first bounce, and the maximum height.
(B) Multiply the time you get in (A) by the horizontal velocity component, 9.92 m/s* cos 75.