Question

1. observe the velocity time graph:

a) what is the slope of the graph?
b) when will the object reach a velocity of -15 m/s?
c) how fast is the object at a time of 15 seconds?
d) when will the object reach a speed of -20 m/s?
e) is the object speeding up or slowing down? how do you know?

Explanation:

Step1: Find the equation of the line

The general form of a linear equation is \(v = mt + b\), where \(m\) is the slope and \(b\) is the \(y -\)intercept. From part (a), \(m=- 2\) (slope). When \(t = 0\), \(v=20\), so \(b = 20\). The equation is \(v=-2t + 20\).

Step2: Solve for \(v\) when \(t = 15\)

Substitute \(t = 15\) into \(v=-2t + 20\).
\(v=-2\times15 + 20\)
\(v=-30 + 20\)
\(v=-10\)

Step3: Solve for \(t\) when \(v=-20\)

Substitute \(v =-20\) into \(v=-2t + 20\).
\(-20=-2t + 20\)
Add \(2t\) to both sides: \(2t-20 = 20\)
Add \(20\) to both sides: \(2t=40\)
Divide by \(2\): \(t = 20\)

Step4: Determine if speeding up or slowing down

The acceleration \(a=m=-2m/s^{2}\) (negative, so decelerating). The velocity is decreasing (since \(m=-2<0\)), so the object is slowing down.

Answer:

c) The object's speed at \(t = 15\) seconds is \(-10m/s\).
d) The object reaches a speed of \(-20m/s\) at \(t = 20\) seconds.
e) The object is slowing down. Because the slope (acceleration) of the \(v - t\) graph is negative (\(a=-2m/s^{2}\)), indicating a decrease in velocity over time.