a rock is thrown straight up in the air from an initial height of ho, in metres, with the initial velocity vo, in metres per second. The height in metres above the ground after t seconds is given by h(t) = -4.9t^2 + vo + ho. Find an expression for the time it takes the rock to reach its maximum height.

this just a parabola, with vertex at

t = -2vo/-9.8

Why did the rock bring a stopwatch to its flight? Because it wanted to measure its peak performance!

To find the time it takes for the rock to reach its maximum height, we need to determine when the velocity of the rock becomes zero. At the maximum height, the velocity is zero since the rock momentarily stops before falling back down.

Since the acceleration due to gravity is -9.8 m/s^2 (negative because it acts in the opposite direction of motion), we can set the velocity equation equal to zero and solve for time:

0 = vo - 9.8t

Solving for t:

9.8t = vo

t = vo / 9.8

So, the expression for the time it takes the rock to reach its maximum height is t = vo / 9.8.

To find the time it takes for the rock to reach its maximum height, we need to determine when the velocity of the rock is zero. When the velocity is zero, the upward motion stops and the rock starts to fall back down.

The velocity of the rock at any time t is given by the derivative of the height function h(t) with respect to time:

v(t) = h'(t) = -9.8t + vo

Setting the velocity equal to zero, we can solve for t:

-9.8t + vo = 0

Solving for t, we get:

t = vo / 9.8

Therefore, the expression for the time it takes the rock to reach its maximum height is t = vo / 9.8.

To find the time it takes for the rock to reach its maximum height, we need to find the value of t when the rock's height is at its maximum.

In the given equation, h(t) = -4.9t^2 + vo + ho, the term -4.9t^2 represents the effect of gravity on the rock's height. This term is negative because gravity pulls the rock downwards.

To find the maximum height, we need to find the maximum value of h(t). The maximum value of a quadratic function occurs at the vertex. The vertex of a quadratic function in the form ax^2 + bx + c is given by the formula x = -b / (2a).

In the equation h(t) = -4.9t^2 + vo + ho, a = -4.9 and b = 0 (since there is no t-term in the equation). Plugging these values into the vertex formula, we get t = -0 / (2 * -4.9) = 0.

This means that the rock reaches its maximum height at t = 0 seconds. Therefore, the expression for the time it takes the rock to reach its maximum height is t = 0 seconds.