On a distant planet, golf is just as popular as it is on earth. A golfer tees off and drives the ball 3.7 times as far as he would have on earth, given the same initial velocities on both planets. The ball is launched at a speed of 44 m/s at an angle of 52° above the horizontal. When the ball lands, it is at the same level as the tee.

To solve this problem, we can use projectile motion equations in both the x and y directions to determine the range of the golf ball on Earth and the distant planet.

Let's start by breaking down the information given:

Initial velocity of the golf ball (v): 44 m/s
Launch angle (θ): 52°
The ball lands at the same level as the tee.

Now, let's calculate the range of the golf ball on Earth.

First, we need to find the horizontal and vertical components of the initial velocity (v_x and v_y).

v_x = v * cos(θ)
v_x = 44 m/s * cos(52°)
v_x = 44 m/s * 0.6157
v_x ≈ 27.07 m/s

v_y = v * sin(θ)
v_y = 44 m/s * sin(52°)
v_y = 44 m/s * 0.7934
v_y ≈ 34.89 m/s

Using the formula for the range (R) in projectile motion: R = (v_x * t)

Since the vertical displacement of the ball is zero (it lands at the same level as the tee), we can use the time of flight (t) to find the range.

Using the formula for time of flight for a projectile launched at an angle: t = (2 * v_y) / g

g = acceleration due to gravity on Earth = 9.8 m/s²

t = (2 * 34.89 m/s) / 9.8 m/s²
t ≈ 7.10 s

R = (27.07 m/s) * (7.10 s)
R ≈ 192.48 m

Now, let's calculate the range of the golf ball on the distant planet using the information given.

Given that the ball travels 3.7 times as far as it would on Earth, we can simply multiply the range on Earth by 3.7.

R_distant_planet = R * 3.7
R_distant_planet ≈ 192.48 m * 3.7
R_distant_planet ≈ 711.62 m

Therefore, on the distant planet, the golf ball's range is approximately 711.62 meters.

To solve this problem, we need to break it down into steps. Let's begin:

Step 1: Calculate the horizontal and vertical components of the initial velocity.
Given:
- Initial speed (vi) = 44 m/s
- Launch angle (θ) = 52°

The horizontal component of velocity (vix) is given by:
vix = vi * cos(θ)

In this case:
vix = 44 m/s * cos(52°)

Step 2: Calculate the vertical component of the initial velocity.
The vertical component of velocity (viy) is given by:
viy = vi * sin(θ)

In this case:
viy = 44 m/s * sin(52°)

Step 3: Calculate the time of flight.
To find the time of flight (t), we can use the equation:
0 = viy * t - 1/2 * g * t^2

Rearranging the equation:
t = (2 * viy) / g

In this case:
t = (2 * (44 m/s * sin(52°))) / 9.8 m/s^2

Step 4: Calculate the total horizontal distance.
The total horizontal distance (d) can be calculated using:
d = vix * t

In this case:
d = (44 m/s * cos(52°)) * t

Step 5: Calculate the distance on the distant planet.
Given:
- The golf ball on the distant planet travels 3.7 times the distance it would on Earth, given the same initial velocities.

We can calculate the distance on Earth (d_earth) by dividing the total horizontal distance by 3.7:
d_earth = d / 3.7

Step 6: Final Answer.
The distance the golfer drove the ball on Earth is d_earth.

Hope this helps! Let me know if you have any further questions.