A stone is thrown vertically upward with a velocity of 58 ft./sec.
A.) How high will it rise?
B.) How soon will it reach the ground again?
C.) What is its velocity half-way up?
A.) Oh, the stone is aiming for the sky! Well, to figure out how high it'll rise, we need to use some physics magic. We can apply the equation:
Final velocity squared = Initial velocity squared + 2 times acceleration times distance
Since the stone is thrown vertically upwards, its final velocity at the top will be 0 ft/sec (because it'll momentarily stop before falling back down). The initial velocity is given as 58 ft/sec, and the acceleration we'll consider is due to gravity, which is approximately 32 ft/sec².
0² = 58² + 2 * (-32) * distance
Now we can solve for distance. By rearranging the equation, we have:
Distance = (0² - 58²) / (2 * -32)
Calculating this, we find the stone will rise approximately 67.94 feet high.
B.) Ah, the inevitable return to the ground! To determine how soon the stone will reach the ground again, we can use a different equation. Considering the initial velocity and acceleration due to gravity, we can use the formula:
Time = (Final velocity - Initial velocity) / acceleration
Since the stone will be falling back down, the initial velocity will be 0 ft/sec, and the final velocity doesn't matter for this calculation. The acceleration remains at approximately 32 ft/sec².
Time = (0 - 58) / (-32)
By plugging in the numbers, we get a time of approximately 1.81 seconds.
C.) Ah, halfway up the stone's journey. It's in an awkward spot, isn't it? Well, let's find its velocity at that point. We can use the equation:
Final velocity = Initial velocity + acceleration * time
Since the stone is halfway up, we'll need to divide the total time it takes to reach the top by 2. In this case, the initial velocity is 58 ft/sec, acceleration is -32 ft/sec², and the time taken will be half of what we calculated earlier.
Final velocity = 58 + (-32) * (1.81 / 2)
After crunching the numbers, we find the velocity halfway up is approximately -21.39 ft/sec. The negative sign indicates that the stone is moving downward at that point. Watch out below!
A.) To find out how high the stone will rise, we can use the equation of motion for vertical motion:
v^2 = u^2 + 2as
where:
v = final velocity (0 ft/sec, as the stone will momentarily come to a stop at its highest point)
u = initial velocity (58 ft/sec)
a = acceleration (in this case, due to gravity, it is -32 ft/sec^2)
s = displacement (height)
Rearranging the equation to solve for s:
s = (v^2 - u^2) / (2a)
Plugging in the values:
s = (0^2 - 58^2) / (2 * (-32))
s = 58^2 / 64
s = 3364 / 64
s ≈ 52.56 ft
Therefore, the stone will rise to a height of approximately 52.56 ft.
B.) To calculate the time it takes for the stone to reach the ground again, we can use the equation of motion:
v = u + at
where:
v = final velocity (0 ft/sec, as the stone will momentarily stop at the ground)
u = initial velocity (58 ft/sec)
a = acceleration (in this case, due to gravity, it is -32 ft/sec^2)
t = time
Rearranging the equation to solve for t:
t = (v - u) / a
Plugging in the values:
t = (0 - 58) / (-32)
t = 58 / 32
t ≈ 1.81 seconds
Therefore, it will take approximately 1.81 seconds for the stone to reach the ground again.
C.) The velocity of the stone halfway up can be found using the equation of motion:
v = u + at
where:
v = final velocity (unknown)
u = initial velocity (58 ft/sec)
a = acceleration (in this case, due to gravity, it is -32 ft/sec^2)
t = time (unknown)
Since we want to find the velocity halfway up, we need to find the time it takes to reach that point. Since the stone is thrown vertically upwards, it will reach its highest point halfway through its total time of flight. Therefore, we can divide the total time of flight by 2 to find the time halfway up.
Total time of flight can be found using the equation:
t = (v - u) / a
where:
v = final velocity (0 ft/sec, as the stone will momentarily stop at its highest point)
u = initial velocity (58 ft/sec)
a = acceleration (in this case, due to gravity, it is -32 ft/sec^2)
Plugging in the values:
t = (0 - 58) / (-32)
t = 1.81 seconds (as calculated earlier)
Therefore, the time halfway up is approximately t/2 = 1.81 / 2 = 0.905 seconds.
Now, we can find the velocity halfway up using the equation of motion:
v = u + at
Plugging in the values:
v = 58 + (-32) * 0.905
v ≈ 58 + (-28.96)
v ≈ 29.04 ft/sec
Therefore, the velocity halfway up is approximately 29.04 ft/sec.
To solve these questions, we'll need to use equations of motion and a basic understanding of physics.
A.) How high will it rise?
To find the height the stone will rise, we can use the equation:
v_f^2 = v_i^2 + 2as
where:
- v_f is the final velocity (which is 0 since the stone reaches its maximum height)
- v_i is the initial velocity (58 ft./sec. upwards)
- a is the acceleration (in this case, acceleration due to gravity, which is -32 ft./sec.^2)
- s is the distance or height we want to find
Rearranging the equation to solve for s, we have:
s = (v_f^2 - v_i^2) / (2a)
Substituting the given values into the equation:
s = (0^2 - 58^2) / (2 * -32)
s = (-3364) / (-64)
s = 52.56 ft.
So, the stone will rise to a height of approximately 52.56 ft.
B.) How soon will it reach the ground again?
To determine the time it takes for the stone to reach the ground, we can use another equation:
v_f = v_i + at
where:
- v_f is the final velocity (which is the speed of the stone just before it hits the ground, which will be negative since it's falling downwards)
- v_i is the initial velocity (58 ft./sec. upwards)
- a is the acceleration (in this case, acceleration due to gravity, which is -32 ft./sec.^2)
- t is the time we want to find
Since the stone reaches the ground again, its final velocity will be the same magnitude as its initial velocity but in the opposite direction. So, we can rewrite the equation as:
-58 = 58 + (-32)t
-58 - 58 = -32t
-116 = -32t
Dividing both sides by -32, we get:
t = 3.625 sec.
Therefore, it will take approximately 3.625 seconds for the stone to hit the ground again.
C.) What is its velocity halfway up?
To determine the velocity of the stone halfway up, we need to find its velocity when it reaches that point. We can do this using the equation of motion again:
v_f = v_i + at
We know that at the halfway point, the stone's velocity will be zero, and we need to find the time it takes to reach there. Let's call this time t_h.
0 = 58 + (-32)t_h
-32t_h = -58
t_h = 1.8125 sec.
Now that we have the time, we can use it to find the velocity halfway up:
v_f_h = v_i + a(t_h)
v_f_h = 58 + (-32)(1.8125)
v_f_h = 58 - 58
v_f_h = 0 ft./sec.
Therefore, the velocity of the stone halfway up will be 0 ft./sec.
oh my, still in feet?
in this case g = about 32 ft/s^2
v = 58 - 32 t
at top t = 0
t at top = 58/32 seconds
total time in air by the way = 2 t = 58/16 seconds
now the problem:
h = Vi t - .5 * 32 t^2
H = 58 (58/32) - .5 * 58^2/32
(A.) H = 58^2/64 ft
(B.) 2 t = 58/16 sec
when h = H/2 = 58^2/128
easy way to do this is with energy
Ke at bottom = (1/2) m(58^2)
energy halfway up
= m g h + .5 m v^2
= 32 m H/2 + .5 m v^2
so
(1/2)(58^2) = 16(58^2/64)+ .5 v^2
solve for v