A golf ball at a mini golf course travels 4.2 m along a carpeted green. When the ball reaches the hole 3.0 s later, its speed is 1.3 m/s. Assuming the ball undergoes constant uniform acceleration, what is the ball's initial speed?
2(4.2)
-------- - 1.3 = 2.8-1.3 = *1.5 m/s*
3
What’s the answer
To find the ball's initial speed, we can use the equation for motion under constant acceleration:
v = u + at
Where:
v = final velocity (1.3 m/s)
u = initial velocity (unknown)
a = acceleration (unknown)
t = time (3.0 s)
Since the ball undergoes uniform acceleration, we can use the equation for displacement:
S = ut + 1/2 at^2
Where:
S = displacement (4.2 m)
u = initial velocity (unknown)
a = acceleration (unknown)
t = time (3.0 s)
We can rearrange this equation to solve for acceleration:
a = 2(S - ut) / t^2
Substituting the given values into the equation:
a = 2 (4.2 - u * 3) / (3^2)
Simplifying, we get:
a = (8.4 - 6u) / 9
Now, we can substitute the values of final velocity and time into the first equation to solve for initial velocity:
1.3 = u + a * 3
Substituting the previously derived value of acceleration:
1.3 = u + ((8.4 - 6u) / 9) * 3
Simplifying, we get:
1.3 = u + (2.8 - 2u) / 3
Multiplying both sides by 3 to eliminate the fraction:
3.9 = 3u + 2.8 - 2u
Combining like terms:
3.9 - 2.8 = 3u - 2u
1.1 = u
Therefore, the ball's initial speed is 1.1 m/s.
To find the ball's initial speed, we can use the equation of motion:
v = u + at
Where:
- v is the final velocity (1.3 m/s)
- u is the initial velocity (which we want to find)
- a is the acceleration (which we'll calculate using another equation)
- t is the time taken (3.0 s)
First, let's calculate the acceleration using the equation:
s = ut + (1/2)at^2
Where:
- s is the distance traveled (4.2 m)
- u is the initial velocity (which we want to find)
- a is the acceleration (which we'll calculate)
- t is the time taken (3.0 s)
Plugging in the given values:
4.2 = u(3.0) + (1/2)a(3.0)^2
Rearranging the equation:
4.2 = 3u + (9/2)a
Now, let's use the equation v = u + at to find the acceleration:
1.3 = u + a(3.0)
Rearranging the equation:
1.3 - 3u = a
Substituting this value for a in the first equation:
4.2 = 3u + (9/2)(1.3 - 3u)
Now we can solve this equation to find the value of u, which will be the initial velocity.