2) Sketch a graph of height vs. time for a projectile fired at 15 degrees, 45 degrees and 60 degrees to the horizontal.

To sketch a graph of height vs. time for a projectile fired at different angles, we can use the equations of motion for projectile motion. The height of the projectile at any given time can be calculated using the equation:

h = h0 + v0sin(θ)t - 0.5gt^2

Where:
- h is the height of the projectile at time t
- h0 is the initial height (usually taken as 0 for simplicity)
- v0 is the initial velocity of the projectile
- θ is the angle of projection
- g is the acceleration due to gravity (approximately 9.8 m/s^2)

To sketch the graph, we need to consider the vertical motion of the projectile. Let's assume the initial velocity, v0, is constant for all three cases. The only variable that will differ among the cases is the angle of projection, θ.

For an object projected at an angle of 0 degrees (vertical projection), the equation becomes:

h = h0 + v0sin(0)t - 0.5gt^2
h = h0 - 0.5gt^2

In this case, the projectile is launched vertically upward, and the height will decrease over time due to the acceleration of gravity. The graphic will show a symmetrical downward parabolic curve.

For an object projected at an angle of 45 degrees (optimal projection), the equation becomes:

h = h0 + v0sin(45)t - 0.5gt^2
h = h0 + (v0/√2) t - 0.5gt^2

In this case, the projectile's initial vertical velocity is equal to v0/√2, giving it a combination of horizontal and vertical velocity components. The height will reach a maximum point and then start to decrease. The graph will show an upward parabolic curve reaching a maximum height and then coming back down symmetrically.

For an object projected at an angle of 60 degrees (near-vertical projection), the equation becomes:

h = h0 + v0sin(60)t - 0.5gt^2
h = h0 + (v0/2) t - 0.5gt^2

In this case, the projectile's initial vertical velocity is equal to v0/2. The height will increase at a slower rate compared to the 45-degree projection and will reach a lower maximum height. The graph will show a less steep upward parabolic curve reaching a lower maximum height and then coming back down symmetrically.

To sketch the graph, you can plot the height (h) on the y-axis and time (t) on the x-axis. Assign values to the initial velocity, v0, and the acceleration due to gravity, g, and then use the equations discussed above to calculate the height at various times. Connect the points to obtain the required height vs. time graph for each angle of projection.