seasoned mini golfer is trying to make par on a tricky hole #5. The golfer can complete the hole by hitting the ball from the flat section it lays on, up a 45° ramp launching the ball into the hole which is d = 1.60 m away from the end of the ramp. If the opening of the hole and the end of the ramp are at the same height, y = 0.690 m, at what speed must the golfer hit the ball to land the ball in the hole? Assume a frictionless surface and the acceleration due to gravity is 9.81 m/s2.
To determine the speed at which the golfer must hit the ball in order to land it in the hole, we can use the principles of projectile motion. Here's how you can calculate it step by step:
Step 1: Split the motion into horizontal and vertical components.
Since there is no friction, the horizontal motion does not affect the vertical motion. We can ignore the horizontal component for now and focus on the vertical motion of the ball.
Step 2: Determine the time of flight.
The time it takes for the ball to travel from the end of the ramp to the hole can be calculated using the formula:
time = sqrt(2 * y / g)
where y is the vertical displacement (0.690 m) and g is the acceleration due to gravity (9.81 m/s^2).
time = sqrt(2 * 0.690 / 9.81)
time ≈ 0.363 s
Step 3: Calculate the vertical component of the initial velocity.
The vertical component of the initial velocity (v₀y) can be determined using the formula:
v₀y = y / time
where y is the vertical displacement (0.690 m) and time is the time of flight (0.363 s).
v₀y = 0.690 / 0.363
v₀y ≈ 1.899 m/s
Step 4: Use the horizontal distance to find the horizontal component of the initial velocity.
Since the horizontal motion is independent of the vertical motion, the horizontal component of the initial velocity (v₀x) remains constant throughout the ball's flight. We can use the horizontal distance (d = 1.60 m) and the time of flight (0.363 s) to calculate it.
v₀x = d / time
v₀x = 1.60 / 0.363
v₀x ≈ 4.407 m/s
Step 5: Find the magnitude of the initial velocity.
The magnitude of the initial velocity (v₀) can be calculated using the horizontal and vertical components found earlier. It can be determined using the equation:
v₀ = sqrt(v₀x^2 + v₀y^2)
v₀ = sqrt((4.407)^2 + (1.899)^2)
v₀ ≈ 4.751 m/s
Therefore, the golfer must hit the ball at a speed of approximately 4.751 m/s to land it in the hole.