You are standing on the ground when a friend standing on a balcony asks you to toss him his keys. He can catch the keys as long as they are at least 4 meters above the ground. If you toss the keys from an initial height of 2 meters with an initial vertical velocity of 1 meter per second, how long does your friend have to catch his keys?

Question

You are standing on the ground when a friend standing on a balcony asks you to toss him his keys. He can catch the keys as long as they are at least 4 meters above the ground. If you toss the keys from an initial height of 2 meters with an initial vertical velocity of 1 meter per second, how long does your friend have to catch his keys?

Answer 1

4 = -4.9 t^2 + 1 t + 2

solve for t ... the time between the two t values is the time above 4 m

using the given values , the keys will never reach 4 m

Answer 2

To determine how long your friend has to catch the keys, we can use kinematic equations of motion. The equation for the vertical displacement of an object in free fall is:

d = vi*t + (1/2)*a*t^2

Where:
d = vertical displacement
vi = initial vertical velocity
a = acceleration due to gravity (g)
t = time

In this case, the initial vertical velocity is 1 m/s, the initial height is 2 meters, and the acceleration due to gravity is approximately 9.8 m/s^2.

We need to find the time it takes for the keys to reach a height of 4 meters, so we set d = 4 meters and solve for t.

4 = 1*t + (1/2)*9.8*t^2

Rearranging the equation:

4 = t + 4.9*t^2

4.9*t^2 + t - 4 = 0

Now we can solve this quadratic equation for t. Using the quadratic formula:

t = (-b +/- sqrt(b^2 - 4*a*c))/(2*a)

In this case, a = 4.9, b = 1, and c = -4.

t = (-1 +/- sqrt(1^2 - 4*4.9*(-4))) / (2 * 4.9)

Simplifying the equation:

t = (-1 +/- sqrt(1 + 78.4)) / (9.8)

t = (-1 +/- sqrt(79.4)) / (9.8)

Calculating the positive root:

t = (-1 + sqrt(79.4)) / (9.8)

t ≈ 0.717 seconds

Therefore, your friend has approximately 0.717 seconds to catch his keys.

Answer 3

To determine how long your friend has to catch the keys, we need to find the time it takes for the keys to reach a height of 4 meters above the ground.

To approach this problem, we can use the concept of projectile motion. When an object is thrown vertically upward, it follows a path described by the equations of motion.

The equation that relates the displacement, initial velocity, time, and acceleration for vertical motion is given by:

s = ut + (1/2)at^2

Where:
s is the displacement (change in height),
u is the initial velocity,
t is the time, and
a is the acceleration due to gravity (approximately -9.8 m/s^2).

Let's substitute the given values into the equation:

s = 4 m (since your friend wants the keys to be at least 4 meters above the ground)
u = 1 m/s (initial vertical velocity)
a = -9.8 m/s^2 (acceleration due to gravity)

4 = 1t - (1/2)(9.8)(t^2)

Now, we can rearrange the equation to solve for time (t):

(1/2)(9.8)(t^2) - 1t + 4 = 0

This is a quadratic equation, so we can solve it using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, the coefficients of the quadratic equation are:
a = (1/2)(9.8) = 4.9
b = -1
c = 4

Plugging these values into the quadratic formula:

t = (-(-1) ± √((-1)^2 - 4(4.9)(4))) / (2(4.9))

Simplifying the equation further:

t = (1 ± √(1 - 78.4)) / 9.8

Now, calculating the values inside the square root:

t = (1 ± √(-77.4)) / 9.8

Since we are taking the square root of a negative number, it means there are no real solutions. Therefore, the keys will not reach a height of 4 meters.