A science project studying catapults sent a projectile into the air with an initial velocity of 24.5 m/s. The formula for distance (s) in meters with respect to time in seconds is
s = -4.9t2 + 24.5t.
Find the time that this projectile would appear to have the maximum distance above the ground.
Calculus? consider the derivative of s wrespect to time, that is vertical velocity, and at max altitude, vertical velocity is zero.
ds/dt=0=-9.8 t+ 24.5 solve for t.
Geometry Method. this is a parabola, s=0 initially, and at the end. Take thequadratic, solve for t when s=0, you get both values (initial and final). at t/2 is the time of max altitude.
To find the time when the projectile appears to have the maximum distance above the ground, you can use the formula for the time of maximum height of a projectile given by t = -b/ (2a), where a and b are the coefficients of the quadratic equation.
In the given equation, s = -4.9t^2 + 24.5t, the coefficient of t^2 is -4.9 and the coefficient of t is 24.5.
Plugging these values into the formula, t = -24.5 / (2*(-4.9)).
Simplifying this expression, t = -24.5 / (-9.8).
Now, divide both the numerator and denominator by -0.1 to simplify further, t = 24.5 / 9.8.
Simplifying this expression, t ≈ 2.5 seconds.
Therefore, the projectile would appear to have the maximum distance above the ground at approximately 2.5 seconds.
To find the time at which the projectile would appear to have the maximum distance above the ground, we need to determine the vertex of the quadratic equation.
In this case, the equation for distance (s) with respect to time (t) is given as:
s = -4.9t^2 + 24.5t
To find the time at which the maximum distance occurs, we can use the formula for the x-coordinate of the vertex of a quadratic equation:
t = -b / (2a)
Comparing the given equation to the standard form of a quadratic equation (ax^2 + bx + c), we can tell that:
a = -4.9
b = 24.5
Substituting these values into the formula for the vertex, we have:
t = -24.5 / (2 * -4.9)
t = -24.5 / -9.8
t ≈ 2.5
Therefore, the projectile would appear to have the maximum distance above the ground at approximately 2.5 seconds.