A golfer is teeing off on a 170 m long par 3 hole. The ball leaves with a velocity of 40 m/s at 50 degrees to the horizontal. Assuming that she hits the ball on a direct path to the hole, how far from the hole will the ball land (no bounces or rolls)?
9.38 m
To find how far from the hole the ball will land, we need to calculate the horizontal distance traveled by the ball.
Step 1: Break down the initial velocity into horizontal and vertical components.
The initial velocity of 40 m/s at an angle of 50 degrees to the horizontal can be divided into two components:
- The horizontal component: v₀x = v₀ * cos(θ)
- The vertical component: v₀y = v₀ * sin(θ)
In this case, v₀ is 40 m/s and θ is 50 degrees.
So, v₀x = 40 * cos(50°)
v₀y = 40 * sin(50°)
Step 2: Determine the time it takes for the ball to reach the ground.
We can use the vertical motion equation to find the time of flight (the time it takes for the ball to reach the ground) since there are no bounces or rolls.
The vertical motion equation is:
y = v₀y * t - (1/2) * g * t²
In this case, the initial vertical position y is 0 (since the ball starts from the ground), and the acceleration due to gravity, g, is approximately 9.8 m/s².
So, the equation becomes:
0 = (40 * sin(50°)) * t - (1/2) * 9.8 * t²
Step 3: Solve for the time of flight.
Rearrange the equation to solve for t, the time of flight:
(1/2) * 9.8 * t² = (40 * sin(50°)) * t
Simplifying further:
4.9 * t² = 19.6 * sin(50°) * t
Divide both sides by t:
4.9 * t = 19.6 * sin(50°)
Now, divide both sides by 4.9:
t = (19.6 * sin(50°)) / 4.9
Using a calculator, you'll find that t ≈ 2.3 seconds.
Step 4: Calculate the horizontal distance traveled.
The horizontal distance traveled, d, can be calculated using the equation:
d = v₀x * t
In this case, v₀x is given by 40 * cos(50°) and t is approximately 2.3 seconds.
So, d = (40 * cos(50°)) * 2.3
Calculating this expression, you'll find that d ≈ 62.3 meters.
Therefore, the ball will land approximately 62.3 meters from the hole.