A Hot Wheels car rolls off a table at a speed of 2.36 m/s and hits the floor a distance of 95.1 cm from the table's edge. Determine the height (in cm) of the table.
Here's a hint: It takes
t = 0.951m/2.36 m/s = 0.404 seconds
to hit the ground.
It falls (g/2)t^2 in time t, because it starts out with zero vertical velocity. Compute that distance.
To find the height of the table, we need to use the kinematic equation:
\[d = v_i t + \frac{1}{2} a t^2\]
Where:
- \(d\) is the distance traveled by the car (95.1 cm),
- \(v_i\) is the initial velocity of the car (2.36 m/s),
- \(a\) is the acceleration due to gravity (-9.8 m/s^2),
- \(t\) is the time it takes for the car to hit the floor (which we need to find).
First, let's convert the initial velocity from m/s to cm/s:
\[v_i = 2.36 \, \text{m/s} \times 100 \, \text{cm/m} = 236 \, \text{cm/s}\]
Now we can plug the values into the equation and solve for \(t\):
\(95.1 \, \text{cm} = 236 \, \text{cm/s} \times t + \frac{1}{2} \times (-9.8 \, \text{m/s}^2) \times t^2\)
Simplifying the equation gives us a quadratic equation:
\[4.9 t^2 - 236t + 95.1 = 0\]
We can solve this equation using the quadratic formula:
\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this case, \(a = 4.9\), \(b = -236\), and \(c = 95.1\).
Plugging in the values:
\[t = \frac{-(-236) \pm \sqrt{(-236)^2 - 4 \times 4.9 \times 95.1}}{2 \times 4.9}\]
Calculating the expression inside the square root:
\[b^2 - 4ac = (-236)^2 - 4 \times 4.9 \times 95.1 = 55504\]
Now, we can continue solving for \(t\):
\[t = \frac{236 \pm \sqrt{55504}}{9.8}\]
\[t \approx \frac{236 \pm 235.72}{9.8}\]
We'll have two solutions, one positive and one negative. Since time cannot be negative, we only consider the positive solution:
\[t \approx \frac{236 + 235.72}{9.8} \approx \frac{471.72}{9.8} \approx 48.13 \, \text{s}\]
Finally, we can use this time to find the height of the table using the following kinematic equation:
\[h = \frac{1}{2} g t^2\]
Where:
- \(h\) is the height of the table,
- \(g\) is the acceleration due to gravity (9.8 m/s^2),
- \(t\) is the time it takes for the car to hit the floor (48.13 s).
Plugging in the values:
\[h = \frac{1}{2} \times 9.8 \, \text{m/s}^2 \times (48.13 \, \text{s})^2\]
Converting the height from meters to centimeters (1 m = 100 cm):
\[h = \frac{1}{2} \times 9.8 \, \text{m/s}^2 \times (48.13 \, \text{s})^2 \times 100 \, \text{cm/m} = 115,829 \, \text{cm}\]
Therefore, the height of the table is approximately 115,829 cm.
To determine the height of the table, we can use the formula for the range of a projectile. The range can be defined as the horizontal distance traveled by the object. In this case, the range is given as 95.1 cm.
The formula to calculate the range is:
Range = (initial velocity^2 * sin(2θ)) / g
In this formula:
- Range is the horizontal distance traveled by the object.
- Initial velocity is the velocity at which the object rolls off the table.
- θ is the launch angle, which is the angle between the horizontal and the initial velocity.
- g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Since the Hot Wheels car rolls off the table horizontally, the launch angle θ is 0 degrees. Therefore, sin(2θ) is equal to sin(0) which is 0.
Let's use the formula and solve for the height (h):
Range = (initial velocity^2 * sin(2θ)) / g
Substituting the given values:
95.1 cm = (2.36 m/s)^2 * 0 / 9.8 m/s^2
Simplifying:
95.1 cm = 5.5536 m^2/s^2 / 9.8 m/s^2
Converting centimeters to meters:
0.951 m = 0.567 m^2/s^2 / 9.8 m/s^2
Rearranging the equation to solve for height (h):
h = initial velocity^2 * (sin(2θ) / g)
Since sin(2θ) is 0, the value of (sin(2θ) / g) is also 0. Therefore, the height of the table is 0 cm.
Hence, the height of the table is 0 cm since the Hot Wheels car rolls off the table horizontally without any initial vertical velocity.