After solving a difficult physics problem, an excited student throws his book straight up. It leaves his hand at 2.5 m/s from 1.5 m above the ground.
A) How much time does it take until the book hits the floor?
B) What's its velocity then?
h = 1.5 + 2.5 t -4.9 t^2
when is h = 0?
solve quadratic for t
4.9 t^2 -2.5 t - 1.5 = 0
you could use http://www.math.com/students/calculators/source/quadratic.htm
Note, quadratics have two solutions of course. In this problem if you get a neagative time, that is how long ago it could have passed the floor on the way up :)
v = Vi - 9.81 t = 2.5 -9.81 t
it better be negative because it is headed down when it hits the floor :)
To find the time it takes for the book to hit the floor, we can use the equation of motion for a vertical free fall:
h = Vi*t + (1/2) * a * t^2,
where:
- h is the height of the book above the ground (1.5 m),
- Vi is the initial vertical velocity of the book (2.5 m/s),
- a is the acceleration due to gravity (-9.8 m/s^2), and
- t is the time in seconds.
A) The book hits the floor when its height (h) becomes zero. So we can rearrange the equation to solve for t:
0 = Vi*t + (1/2) * a * t^2.
This is a quadratic equation. Rearranging further, we get:
(1/2) * a * t^2 + Vi*t = 0.
Multiplying through by 2 to get rid of the fraction:
a * t^2 + 2 * Vi * t = 0.
Now, we can solve this equation using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac))/(2a),
where a = a, b = 2 * Vi, and c = 0.
Plugging in these values:
t = (-2 * Vi ± sqrt((2 * Vi)^2 - 4 * a * 0))/(2 * a).
To find the correct solution, we need to choose the positive value for time (t > 0) since we're interested in the time it takes for the book to hit the ground.
B) To find the velocity of the book at the instant it hits the floor, we can use the equation:
Vf = Vi + a*t,
where Vf is the final velocity of the book just before it hits the ground.
Let's calculate both A) and B) step by step.
A) Calculating the time it takes for the book to hit the floor:
- Substitute the values of Vi, a, and c into the quadratic formula:
t = (-2 * 2.5 ± sqrt((2 * 2.5)^2 - 4 * (-9.8) * 0))/(2 * (-9.8)).
- Simplifying further:
t = (-5 ± sqrt(25 + 0))/(-19.6).
- Calculating the square root and final values:
t = (-5 ± 5)/(-19.6), which gives two possible values for t.
- Choosing the positive value:
t = (5 - 5)/(-19.6) or t = (5 + 5)/(-19.6).
- Simplifying:
t = 0/(-19.6) or t = 10/(-19.6).
- Thus, the time it takes for the book to hit the floor is t = 0 seconds or t ≈ 0.51 seconds.
B) Calculating the velocity of the book at the instant it hits the floor:
- Substitute the values of Vi, a, and t into the equation:
Vf = 2.5 + (-9.8) * 0.51.
- Simplifying further:
Vf ≈ 2.5 - 4.998.
- Thus, the velocity of the book at the instant it hits the floor is approximately Vf ≈ -2.498 m/s (downward direction).
Note: The negative sign indicates that the velocity is in the downward direction.
A) To find the time it takes for the book to hit the floor, we can use the kinematic equation for the vertical motion:
𝑑 = 𝑣₀𝑡 + (1/2)𝑎𝑡²
where:
- 𝑑 is the vertical displacement (from 1.5 m above the ground to the ground, so 𝑑 = -1.5 m),
- 𝑣₀ is the initial velocity (2.5 m/s, but in the opposite direction since the book is thrown upwards, so 𝑣₀ = -2.5 m/s),
- 𝑎 is the acceleration (due to gravity, 𝑎 = -9.8 m/s²),
- 𝑡 is the time.
By substituting the values into the equation, we have:
-1.5 = -2.5𝑡 + (1/2)(-9.8)(𝑡)²
We can rearrange the equation to solve for 𝑡:
-1.5 = -2.5𝑡 - 4.9𝑡²
Combining like terms, we get:
4.9𝑡² + 2.5𝑡 - 1.5 = 0
Solving this quadratic equation, we find that 𝑡 ≈ 0.49 s.
Therefore, it takes approximately 0.49 seconds for the book to hit the floor.
B) We can find the velocity of the book at the instant it hits the floor by using the kinematic equation:
𝑣 = 𝑣₀ + 𝑎𝑡
where:
- 𝑣 is the final velocity (what we want to find),
- 𝑣₀ is the initial velocity (-2.5 m/s),
- 𝑎 is the acceleration due to gravity (-9.8 m/s²),
- 𝑡 is the time (0.49 s).
Substituting the given values into the equation, we have:
𝑣 = -2.5 + (-9.8)(0.49)
Simplifying the equation, we get:
𝑣 ≈ -2.5 - 4.802
Therefore, the velocity of the book when it hits the floor is approximately -7.302 m/s (since the negative sign indicates it is moving downward).