The Problem:
A cheetah is pursuing an impala. The impala is running in a straight line at a constant speed of 16 m/s. The cheetah is 10 m behind the impala, running at 20 m/s but tiring, so that it is decelerating at 1 ms -2. Find an expression for the gap between the cheetah and the impala t seconds later. Will the impala get away?
A cheetah is pursuing an impala. The impala is running in a straight line at a constant speed of 16 m/s. The cheetah is 10 m behind the impala, running at 20 m/s but tiring, so that it is decelerating at 1 m/s². Find an expression for the gap between the cheetah and the impala t seconds later. Will the impala get away?
Solution:
Step 1: Position of the Impala
The impala is moving at a constant speed, so its position at time t is:
x_i(t) = 16t
Step 2: Position of the Cheetah
The cheetah is decelerating, so its position at time t is:
x_c(t) = -10 + 20t - (1/2) t²
Step 3: Gap Between the Cheetah and the Impala
The gap at time t is:
G(t) = x_i(t) - x_c(t)
Substitute the expressions for x_i(t) and x_c(t):
G(t) = 16t - (-10 + 20t - (1/2) t²)
Simplify:
G(t) = (1/2) t² - 4t + 10
Step 4: Will the Impala Escape?
The gap G(t) is a quadratic function that opens upwards (since the coefficient of t² is positive). This means the gap will increase indefinitely over time, and the impala will escape.
Final Answer:
- Expression for the gap:
G(t) = (1/2) t² - 4t + 10 - Will the impala escape? Yes, the impala will escape.
